Mathematics High School

## Answers

**Answer 1**

It is essential to **analyze** and interpret the graphs or **equations** provided to make accurate comparisons and draw meaningful conclusions.

To find the slope of one section of the displacement plot, we need to identify a specific portion of the displacement curve and calculate the slope by taking the change in **displacement** over the corresponding time interval.

The average velocity during the same time interval can be found by dividing the total change in displacement by the total time elapsed.

Comparing the **slope** of the displacement curve to the corresponding average velocity value allows us to observe how the instantaneous rate of change (slope) compares to the overall average rate of change (average velocity) over the same time interval.

Comparing the change in position to the area under the velocity curve involves calculating the total area under the velocity curve for the given time interval and comparing it to the total change in position during that time interval. This allows us to see if the overall displacement matches the total area under the velocity curve.

Similarly, comparing the change in velocity to the area under the acceleration curve involves calculating the total area under the acceleration curve for the given time interval and comparing it to the total change in velocity during that time interval. This helps us determine if the change in velocity corresponds to the total effect of the acceleration over the given time interval.

The specific values and comparisons will depend on the specific context and the given displacement, velocity, and acceleration curves. It is essential to analyze and interpret the graphs or equations provided to make accurate comparisons and draw meaningful conclusions.

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## Related Questions

Given a = 11, b = 20, and c = 11, use the Law of Cosines to find angle B. Round to three decimal places.

1. 24.620⁰ 2. 64.240° 3. 30.670⁰

4. 130.760°

### Answers

By using the **Law of** **Cosines**, the value of angle B is 130.760°. The option 4 is correct answer.

The Law of Cosines is a** fundamental theorem** in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is an extension of the Pythagorean theorem and can be used to solve triangles in cases where the lengths of all three sides or two sides and the included angle are known.

The Law of Cosines states:

b² = a² + c² - 2ac × cos(B)

where:

b is the **length** of the side opposite angle B.

a and c are the lengths of the other two sides of the triangle.

B is the angle opposite side b.

Now, let's substitute the values:

a = 11

b = 20

c = 11

we can substitute** **these values into the **equation**:

20² = 11² + 11² - 2 × 11 × 11 × cos(B)

400 = 121 + 121 - 242 × cos(B)

400 = 242 - 242 × cos(B)

242 × cos(B) = 242 - 400

242 × cos(B) = -158

cos(B) = -158 / 242

B = arccos(-158 / 242)

Rounding to three **decimal **places, B ≈ 130.760°.

Therefore, the correct answer is option 4: 130.760°.

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Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(X)= -2 Var(X) = 21 E(Y)=-4 E (Z)=-7 Var(Y) = 30 Var (Z)= 18 Compute the values of the expressions below. E(-52-1)= 0 x Х ? E (**»2-0 X+3Z 5 Var(-4Y)-2= 0 E(-4x2)= 0

### Answers

The values of the **expressions **are as follows:

E(-52 - 1) = -53

0 x Х = 0

E(2X - 0 + 3Z) = -25

Var(-4Y - 2) = 480

E(-4X^2) = -100

To compute the values of the given expressions, let's use the properties of expectation and **variance.**

E(-52 - 1):

We can distribute the expectation operator over addition and constants, so E(-52 - 1) = E(-52) - E(1).

Since -52 and 1 are constants, their **expectations **are equal to themselves. Therefore, E(-52) - E(1) = -52 - 1 = -53.

0 x Х:

Multiplying any random variable by zero results in zero. So, 0 x Х = 0.

E(2X - 0 + 3Z):

Using linearity of expectation, E(2X - 0 + 3Z) = 2E(X) - E(0) + 3E(Z).

Since the expected value of a **constant i**s the constant itself, E(0) = 0.

Substituting the given values, 2E(X) - E(0) + 3E(Z) = 2(-2) + 0 + 3(-7) = -4 - 21 = -25.

Var(-4Y - 2):

Using the properties of variance, Var(-4Y - 2) = Var(-4Y) = (-4)^2 Var(Y).

Substituting the given variance of Y, Var(-4Y) = 16 * 30 = 480.

E(-4X^2):

Applying the constant factor rule for expectation, E(-4X^2) = -4E(X^2).

To find E(X^2), we need to use the property Var(X) = E(X^2) - [E(X)]^2. We know Var(X) = 21 and E(X) = -2.

**Rearranging **the equation, E(X^2) = Var(X) + [E(X)]^2 = 21 + (-2)^2 = 21 + 4 = 25.

Substituting the value into E(-4X^2), we get -4 * 25 = -100.

Therefore, the values of the expressions are as follows:

E(-52 - 1) = -53

0 x Х = 0

E(2X - 0 + 3Z) = -25

Var(-4Y - 2) = 480

E(-4X^2) = -100

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If sin(∅)12/13, 0 ≤ ∅≤ π/2 then cos(∅) equals

tan (∅) equals sec (∅) equals Round answers to 3 decimal places If cos(∅) = 10,0 ≤ ∅≤π/2 then sin(∅) equals tan (∅) equals

sec (∅) equals

Round answers to 3 decimal places

If cos(∅) = 5/3 ,0 ≤∅ ≤ π/2then sin(∅) equals tan (∅) equals

sec (∅) equals

give an exact answer ( no rounded decimals)

If cos(∅) = 15/8,0 ≤ ∅≤ π/2 then sin(∅) equals tan (∅) equals

sec (∅) equals

### Answers

The **values **are as follows:

1) For sin(∅) = 12/13, 0 ≤ ∅ ≤ π/2: cos(∅) = 5/13, tan(∅) = 12/5, sec(∅) = 13/5.

2) For cos(∅) = 10, 0 ≤ ∅ ≤ π/2: sin(∅) is undefined, tan(∅) is undefined, sec(∅) = 1/10.

What are the values of cos(∅), tan(∅), and sec(∅) given certain conditions?

1) If sin(∅) = 12/13, 0 ≤ ∅ ≤ π/2:

cos(∅) = 5/13 tan(∅) = 12/5 sec(∅) = 13/5

2) If cos(∅) = 10, 0 ≤ ∅ ≤ π/2:

sin(∅) is **undefined **(since |sin(∅)| ≤ 1) tan(∅) is undefined (since cos(∅) = 10 implies sin(∅) = √(1 - cos²(∅)) is not a real number)sec(∅) = 1/10

3) If cos(∅) = 5/3, 0 ≤ ∅ ≤ π/2:

sin(∅) = √(1 - cos²(∅)) = 4/3tan(∅) = sin(∅)/cos(∅) = 4/5 sec(∅) = 3/5

4) If** cos(∅)** = 15/8, 0 ≤ ∅ ≤ π/2:

sin(∅) = √(1 - cos²(∅)) = √(1 - (15/8)²) = √(1 - 225/64) = √(64/64 - 225/64) = √(-161/64) (exact answer)tan(∅) = sin(∅)/cos(∅) = √(-161/64) / (15/8) = -8√(161)/15 (exact answer)sec(∅) = 8/15 (exact answer)

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Find the solution to the linear system of differential equations y' y(0) = 3. x(t) = y(t) = = = -15x - 8y 24x + 13y satisfying the initial conditions x(0) = −3 and y(0) = 3

### Answers

The solution to the given linear system of** differential equations** is x(t) = -3e^(-7t) - 3e^(6t) and y(t) = 3e^(-7t) - e^(6t), satisfying the initial conditions x(0) = -3 and y(0) = 3.

The **solution **can be obtained by solving the system of differential equations. From the given equations, we have **dx/dt = -15x - 8y** and dy/dt = 24x + 13y. By rearranging the equations, we get dx/dt + 15x + 8y = 0 and dy/dt - 24x - 13y = 0.

Solving these two **equations**, we find that x(t) = -3e^(-7t) - 3e^(6t) and y(t) = 3e^(-7t) - e^(6t). These solutions satisfy the given initial conditions x(0) = -3 and y(0) = 3, which means that when t = 0, x is -3 and y is 3.

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3). Assume a normally distributed population with μ = 80 and a=5 Using Appendix C-1 What proportion of scores in this distribution is equal to or greater than 88? What proportion of scores in this distribution is between 83 and 87

### Answers

In this scenario, we are assuming a normally distributed population with a mean (μ) of 80 and a** standard deviation** (σ) of 5. We want to determine the proportion of scores in this distribution that are equal to or greater than 88, as well as the proportion of scores between 83 and 87. By referring to Appendix C-1 or using statistical software, we can calculate the desired **proportions**.

To find the proportion of scores in the distribution that are equal to or greater than 88, we need to calculate the **z-score** corresponding to the value 88 and then find the **cumulative probability** associated with that z-score. The z-score formula is given by:

z = (x - μ) / σ

where x is the value of interest, μ is the mean, and σ is the standard deviation.

Substituting the given values:

z = (88 - 80) / 5

Calculating this expression gives us the z-score. By referring to Appendix C-1 or using a z-table or statistical software, we can find the cumulative probability associated with this z-score.

This probability represents the proportion of scores in the distribution that are equal to or greater than 88.

Similarly, to find the proportion of scores between 83 and 87, we need to calculate the z-scores corresponding to these values and then find the difference in cumulative probabilities between the two z-scores. By using the z-score formula and the given values, we can calculate the z-scores for 83 and 87.

Then, by referring to Appendix C-1 or using a z-table or statistical software, we can find the cumulative probabilities associated with these z-scores. The** difference** between these probabilities represents the proportion of scores in the distribution that are between 83 and 87.

It's important to note that Appendix C-1 provides a table of cumulative probabilities for different z-scores. By locating the z-score calculated for each case and finding the corresponding cumulative probability, we can determine the proportions of interest.

Using these calculations, we can find the proportion of scores in the distribution that are equal to or greater than 88 and the proportion of scores between 83 and 87. The specific values can be obtained by referring to Appendix C-1 or using statistical software that provides cumulative probability calculations based on the **normal distribution**.

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iConsider the following two-period model of the current account: U = (1 − 3)In(C₁) + Bln (C₂) C₁ = Y₁ - CA₁, C₂=Y₂+(1+r)СA₁ CA₁+ CA₂ = 0 where C is consumption, CA is the current account balance, and r is the given world interest rate. Y₁, Y₂ > 0 are given endowments in periods 1 and 2 and 0 < 3 < 1 is a known parameter. 1+r 1+r (a) Derive the lifetime budget constraint C₁+₁=Y₁+2 and find analytical solutions for C₁, C2, CA₁, CA2. Show that the home country runs a current account deficit in period 1 if and only if rª >r, where rª is the autarky interest rate. [10%]

### Answers

In the two-period model of the **current account**, the home country runs a current account deficit in period 1 if and only if rª > r.

The lifetime budget constraint in the two-period model is given by C₁/(1+r) + C₂ = Y₁/(1+r) + Y₂, where C₁ and C₂ represent **consumption **in periods 1 and 2, Y₁ and Y₂ are endowments in the respective periods, and r is the world interest rate.

To find analytical solutions for C₁, C₂, CA₁, and CA₂, we can substitute the expressions for C₁ and C₂ from the given equations into the budget constraint. This yields:

(Y₁ - CA₁)/(1+r) + C₂ = Y₁/(1+r) + Y₂.

**Simplifying **the equation, we find:

CA₁ = Y₁ - Y₁/(1+r) + Y₂ - C₂(1+r).

We can further simplify this expression to:

CA₁ = (rªY₁ - Y₁ + Y₂ - C₂(1+r))/(1+r),

where rª is the autarky interest rate.

Now, we observe that the home country runs a current account **deficit **in period 1 if and only if CA₁ is negative, meaning CA₁ < 0. Thus, we have:

rªY₁ - Y₁ + Y₂ - C₂(1+r) < 0.

Rearranging the inequality, we get:

rª > (Y₁ + Y₂ - C₂(1+r))/Y₁.

Therefore, the home country runs a current account deficit in period 1 if and only if rª > r, where rª is the **autarky interest rate**. This condition indicates that the home country prefers **borrowing **from abroad (running a deficit) when the world interest rate is lower than its autarky interest rate.

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Use the diagram below and process introduced in this lesson to

derive the area formula for a triangle in terms of the variables a,

b, and the angle C.∗

*Make sure to use the multiplication symbol

### Answers

To derive the** area formula **for a triangle in terms of the variables a, b, and the angle C, we can start with the standard formula for the area of a triangle:

Area = (1/2) * base * height

In the given diagram, let's consider side a as the base and draw a perpendicular line from the opposite vertex to side a, forming a right triangle. The length of this **perpendicular** line is the height of the triangle.

Now, let's label the vertices of the triangle as follows:

A: Vertex opposite to side a

B: Vertex adjacent to side a

C: Vertex adjacent to side b

Using the given labeling, angle C is opposite to side a, and we have a right triangle with angle C as one of its acute angles.

The length of the perpendicular line (height) from vertex A to side a is given by:

height = b * sin(C)

Substituting this height into the area formula, we have:

Area = (1/2) * a * (b * sin(C))

Simplifying further, we get:

Area = (1/2) * a * b * sin(C)

Therefore, the area formula for a triangle in terms of the** variables **a, b, and the angle C is:

Area = (1/2) * a * b * sin(C)

Note: This derivation assumes that angle C is one of the acute angles of the triangle. If angle C is the obtuse angle, we would use the sine of the supplementary angle (180° - C) instead.

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determine the diameter, to the nearest inch, of a large can of tuna fish that has a volume of 66 cubic inches and a height of 3.3 inches.

### Answers

The **diameter **of the large can of tuna fish, to the **nearest **inch, is 5 inches.

To determine the **diameter **of the can, we need to use the formula for the volume of a cylinder, which is given by V = [tex]\pi r^2h[/tex], where V is the volume, r is the radius, and h is the height. Since we are given the volume V and the height h, we can rearrange the formula to solve for the radius r: r = sqrt(V / (πh)).

Substituting the given **values**, we have r = sqrt(66 / (π 3.3)). Using a calculator, we can calculate the value of r to be approximately 2.978 inches.

Since the diameter is equal to twice the **radius**, we can multiply the radius by 2 to get the diameter: diameter = 2 × 2.978 = 5.956 inches. Rounding to the nearest inch, the diameter of the large can of tuna fish is **approximately **5 inches.

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Find the matrix A of the linear transformation T(f(t))=f(−4) from P 2

to P 2

with respect to the standard basis for P 2

,{1,t,t 2

} A= Note: You should be viewing the transformation as mapping to constant polynomials rather than real numbers, e.g. T(2+t−t 2

)=−4+0t+0t 2

### Answers

To find the matrix A of the **linear **transformation T(f(t)) = f(-4) from P2 to P2 with respect to the standard basis {1, t, t^2}, we need to determine the images of the basis **vectors **under the transformation.

Let's consider each basis vector:

T(1) = 1(-4) + 0t + 0t^2 = -4

The image of the first basis vector is -4, which can be represented as [-4, 0, 0] in the standard basis.

T(t) = t(-4) + 0t + 0t^2 = -4t

The image of the second basis vector is -4t, which can be represented as [0, -4, 0] in the **standard **basis.

T(t^2) = t^2(-4) + 0t + 0t^2 = -4t^2

The image of the third basis vector is -4t^2, which can be represented as [0, 0, -4] in the standard basis.

Therefore, the matrix A representing the linear transformation T is:

A = [[-4, 0, 0],

[0, -4, 0],

[0, 0, -4]]

This matrix represents the **coefficients **of the transformed **polynomials **in the standard basis {1, t, t^2}.

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a rectangle is drawn so the width is 23 inches longer than the height. if the rectangle's diagonal measurement is 65 inches, find the height. give your answer rounded to 1 decimal place.

### Answers

The height of the** rectangle** is 36 inches

Let the** height **of the rectangle be x. Then, the width of the rectangle will be 23 more than the height, i.e. (x + 23).Using the Pythagorean Theorem, we know that for a rectangle with height x and width (x+23), the diagonal of the rectangle, d can be given as:

d² = x² + (x + 23)²d² = x² + x² + 46x + 529d² = 2x² + 46x + 529

Since we are given that the diagonal measurement is 65 inches, we can plug this into our equation to obtain:65² = 2x² + 46x + 5294225 = 2x² + 46x + 5292x² + 46x - 4296 = 0Dividing by 2: x² + 23x - 2148 = 0

Factoring the quadratic equation gives:(x-36)(x+59) = 0Taking x = 36 (since x cannot be negative), the height of the rectangle is 36 inches.

Therefore, the **width** of the rectangle is (36 + 23) = 59 inches. Thus, the height of the rectangle is 36 inches when the width is 59 inches.

The answer is 36.0, rounded to 1 decimal place.

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Given a = 8, b = 12, and c = 6, use the Law of Cosines to find angle C. Round to three decimal places.

1. 36.336°

2. 117.280⁰

3. 26.384°

4. 20.901⁰

### Answers

Angle C, when using the **Law of Cosines** with the given values of a = 8, b = 12, and c = 6, is approximately 26.384°.

Summary: By applying the Law of Cosines to the given triangle, we can determine the **measure **of angle C. Using the formula cos(C) = (a^2 + b^2 - c^2) / (2ab) and substituting the provided values, we calculate cos(C) ≈ 0.896. Taking the **inverse **cosine of 0.896 yields C ≈ 26.384°. Therefore, the correct answer is option 3: 26.384°.

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In Exercises 37, establish the identity. 37. cos ( π/2 + x) = - sin x

### Answers

The value of the **identity **cos(π/2 + x) = -sin(x). The sum formula for cosine is

cos(π/2 + x) = cos(π/2)cos(x) - sin(π/2)sin(x)

To establish the identity cos(π/2 + x) = -sin(x), we can use the sum formula for cosine and the definition of **sine**.

Using the sum formula for **cosine**, we have:

cos(π/2 + x) = cos(π/2)cos(x) - sin(π/2)sin(x)

Now, we know that cos(π/2) = 0 and sin(π/2) = 1, so we can **substitute **these values:

cos(π/2 + x) = 0 * cos(x) - 1 * sin(x)

**Simplifying **further, we get:

cos(π/2 + x) = -sin(x)

Therefore, we have established the identity cos(π/2 + x) = -sin(x).

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a) Find the area enclosed between the curve y = x(x-1)2 and the axis y = 0, establishing first where they intersect. b) Illustrate the extent of the enclosed area by plotting the curve over an appropriate domain and shading the enclosed area.

### Answers

(a) 1/10 square units is the area enclosed between the **curve** y = x(x-1)² and the axis y = 0, establishing first where they intersect. (b) The curve has been plotted and the area has been shaded that is enclosed between y = x(x-1)² and y = 0.

(a) The** **curve y = x(x-1)² intersects the x-axis at x = 0 and x = 1. The area enclosed between the curve and the x-axis can be found by integrating the absolute value of the curve** function** over the interval [0, 1].

To find the points of intersection between the curve y = x(x-1)² and the y-axis, we set y = 0 and solve for x:

x(x-1)² = 0

This equation is satisfied when x = 0 or x = 1. Therefore, the curve intersects the y-axis at these two points.

To calculate the area **enclosed** between the curve and the y-axis, we integrate the absolute value of the function over the interval [0, 1]. Since the curve lies above the x-axis on this interval, we have:

Area = ∫[0, 1] |x(x-1)²| dx

Evaluating the **integral**, we find:

Area = 1/10 square units

(b) To illustrate this enclosed area, we can plot the curve y = x(x-1)² over the **domain** [0, 1]. The curve will be above the x-axis, and we can shade the region between the curve and the x-axis to indicate the enclosed area.

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A researcher compares the effectiveness of two different instructional methods for teaching physiology A sample of 203 students using Method 1 produces a testing avenge of 79 A sample of 244 students in Method producting 738 Au the population standard deviation for Method 1 12.18, while the population standard deviation for Michod 25.51. Demine the confidence for the true difference between testing averages for students using Method 1 and students using Method 2 Step 1 of 2 Find the critical value that should be used in constructing the confidence interval A researcher compares the effectiveness of two different instructional methods for teaching physiology. A sample of 203 students using Method 1 produces a testing average of 79. A sample of 244 students using Method 2 produces a testing average of 73.8. Assume that the population standard deviation for Method 1 is 12.18, while the population standard deviation for Method 2 is 5.51. Determine the 98 % confidence interval for the true difference between testing averages for students using Method 1 and students using Method 2. Step 2 of 2: Construct the 98 % confidence interval. Round your answers to one decimal place. AnswerHow to enter your answer (opens in new window) 2 Points Keypad Keyboard Shortcuts Lower endpoint: Upper endpoint

### Answers

To determine the 98% **confidence interval **for the true difference between testing averages for students using Method 1 and Method 2, we can follow these steps:

Step 1: Find the critical value

Since the sample sizes are large (n1 = 203, n2 = 244), we can assume that the sampling distributions of the means are approximately normal. We'll use the **Z-distribution** to find the critical value for a 98% confidence level.

The critical value for a 98% confidence level can be found using a standard normal distribution table or a calculator. The corresponding Z-score is approximately 2.33.

Step 2: Construct the confidence interval

The formula for the confidence interval for the difference between two **population** means is:

CI = (X1 - X2) ± Z * sqrt((s1^2 / n1) + (s2^2 / n2))

Where:

CI = Confidence Interval

X1 = Sample mean of Method 1

X2 = Sample mean of Method 2

Z = Critical value

s1 = Population standard deviation of Method 1

s2 = Population **standard deviation **of Method 2

n1 = Sample size of Method 1

n2 = Sample size of Method 2

Plugging in the given values:

X1 = 79

X2 = 73.8

Z = 2.33 (for 98% confidence level)

s1 = 12.18

s2 = 5.51

n1 = 203

n2 = 244

CI = (79 - 73.8) ± 2.33 * sqrt((12.18^2 / 203) + (5.51^2 / 244))

Calculating the values inside the square root:

CI = 5.2 ± 2.33 * sqrt(0.7086 + 0.157)

Calculating the square root:

CI = 5.2 ± 2.33 * sqrt(0.8656)

CI = 5.2 ± 2.33 * 0.9307

CI = 5.2 ± 2.1686

CI ≈ (2.03, 8.37)

Therefore, the 98% **confidence interval** for the true difference between testing averages for students using Method 1 and students using Method 2 is approximately (2.0, 8.4).

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Show all work, use exact values.

13) Suppose you have been assigned the job of measuring the height of the local water tower. Climbing makes you dizzy, so you decide to do the whole job at ground level. From a point of 47.3 meters from the base of the tower, you must look up at an angle of 53° to see the top of the tower. How tall is the tower? (5 pts) 14) The CN Tower in Toronto, Ontario is 552 meters tall. At a certain time of day, it casts a shadow of 1100 meters onto the ground. What is the angle of elevation of the sun at that time of day? (5 pts)

### Answers

The height of the water tower, we can use **trigonometry**. Given the distance from the base of the tower (47.3 meters) and the angle of elevation (53°) needed to see the top of the tower, we can calculate the height of the tower.

The angle of **elevation **of the sun, we can use the concept of similar triangles. By comparing the height of the CN Tower (552 meters) with the length of its shadow (1100 meters), we can calculate the angle of elevation of the sun.

13) We can start by drawing a right triangle, where the height of the water tower is the vertical side, the distance from the base of the tower is the base, and the angle of elevation is the angle between the base and the **hypotenuse**.

Let's label the height of the tower as 'h', the distance from the base as 'd', and the angle of elevation as 'θ'.

Using trigonometry, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the tangent of the angle of elevation is equal to the **height **of the tower divided by the distance from the base:

tan(θ) = h/d

the height of the tower, we rearrange the **equation**:

h = d * tan(θ)

Substituting the given values, we have:

h = 47.3 meters * tan(53°)

Using a scientific **calculator**, we can find the value of tan(53°) ≈ 1.32704482162.

Therefore, the height of the water tower is:

h ≈ 47.3 meters * 1.32704482162 ≈ 62.705 meters.

Hence, the tower is approximately 62.705 meters tall.

The angle of elevation of the sun, we can set up a proportion using similar triangles.

Let's label the height of the CN Tower as 'H', the length of its shadow as 'S', and the angle of elevation of the sun as 'θ'.

According to similar triangles, the ratio of the height of the tower to the length of its shadow is equal to the ratio of the height of the sun to the distance between the tower and the sun:

H/S = height of the sun / distance between the tower and the sun

We know the height of the CN Tower is 552 meters, and the length of its shadow is 1100 meters.

Therefore, we have:

552/1100 = height of the sun / distance between the tower and the sun

Simplifying the equation, we get:

0.5 = height of the sun / distance between the tower and the sun

The angle of elevation of the sun, we can take the inverse tangent (arctan) of both sides of the equation:

arctan(0.5) = arctan(height of the sun / distance between the tower and the sun)

Using a scientific calculator, we find that arctan(0.5) ≈ 26.565°.

Hence, the angle of elevation of the sun at that time of day is approximately 26.565°.

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rewrite 7x 49 using a common factor. 7(x 7) 7(x 49) 7x(x 7) 7x(x 49)

### Answers

The correct option is 7(x 7) by **highest common factor.**

The given expression is 7x49. We have to rewrite the given expression using a** common factor**.

To rewrite 7x 49 using a common factor, we can **factor out** 7.

7x49 can be written as 7 × x × 7 × 7.7x49 = 7 × x × 7 × 7

Therefore, the correct option is 7(x 7).

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CHOOSING CARDS A card is randomly selected from a standard deck of 52 cards. Find the probability of drawing the given card.

20. A king and a diamond

21. A king or a diamond

22. A spade or a club

23. A 4 or a 5

24. A 6 and a face card

25. Not a heart

### Answers

To find the probability of drawing a specific card from a standard deck of 52 cards, we divide the number of favorable **outcomes** (cards that match the given criteria) by the total number of possible outcomes (total number of cards in the deck).

A king and a **diamond**:

There is only one king of diamonds in the deck, so the probability of drawing a king and a diamond is 1/52.

A king or a diamond:

There are 4 kings and 13 diamonds in the deck (including the king of diamonds counted only once). However, we need to subtract the king of diamonds once since it was already counted. So, the **probability** of drawing a king or a diamond is (4 + 13 - 1) / 52 = 16/52 = 4/13.

A spade or a club:

There are 13 spades and 13 clubs in the deck. The probability of drawing a spade or a club is (13 + 13) / 52 = 26/52 = 1/2.

A 4 or a 5:

There are 4 fours and 4 fives in the deck. The probability of **drawing** a 4 or a 5 is (4 + 4) / 52 = 8/52 = 2/13.

A 6 and a face **card**:

There are 4 sixes and 12 face cards (3 face cards in each suit: jack, queen, and king). The probability of drawing a 6 and a face card is (4 * 12) / 52 = 48/52 = 12/13.

Not a heart:

There are 52 cards in the deck, and 13 of them are hearts. The probability of drawing a card that is not a heart is (52 - 13) / 52 = 39/52 = 3/4.

Note: The probabilities given here assume that the cards are drawn randomly and without replacement, meaning once a card is drawn, it is not put back into the deck.

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let R be the region in the first quadrant bounded above by y 4x +3 and below by y = x 2 + 3.

(a) Find the area of R

(b) The line x c divides R into two regions. If the area of region R to the area of the region, find the value of c. the left of x c is (c) The line y a divides R into two regions of equal area. Find the value ofa

### Answers

a.The required **area** of R is 32/3 units².

b. Required **value** of c is 3.

c. Required value of a is 13/3.

Given: Let R be the **region** in the first **quadrant** bounded above by y = 4x + 3 and below by y = x² + 3.

We need to find the following.

(a) The area of R.(b) The line xc divides R into two regions. If the area of region R to the area of the region, find the value of c. The left of xc is the line ya divides R into two regions of equal area. Find the value of a.

(a) The area of RThe region R is bounded by the curves y = 4x + 3 and y = x² + 3.First, we find the **x-coordinates** of the points where they intersect.x² + 3 = 4x + 3x² - 4x = 0 and x(x - 4) = 0 and x = 0 or x = 4

So the curves intersect at (0, 3) and (4, 19).

The area of R is given by the integral of the curve y = 4x + 3 minus the integral of the curve y = x² + 3 within the limits of x = 0 and x = 4.∫[0, 4] (4x + 3) dx - ∫[0, 4] (x² + 3) dx[2x² + 3x]0¹⁰ - [1/3 x³ + 3x]0¹⁰ = 32/3 units²

(b) The line xc divides R into two regions. If the area of region R to the area of the region. The left of xc is the area of R is 32/3 units². Let the left region be R1 and the right region be R2. Let the x-coordinate of the point of **intersection** of the line xc with the curve y = 4x + 3 be h.So the equation of the line xc is x = h.Since R1 and R2 have equal areas, they have areas of 16/3 units² each.

Therefore, the left boundary of R2 is x = 2h/3.Using the same method of finding areas as in part (a),∫[0, h] (4x + 3) dx - ∫[0, h] (x² + 3) dx = 16/3 units² and∫[0, 2h/3] (4x + 3) dx - ∫[0, 2h/3] (x² + 3) dx = 16/3 units²

Solving these equations for h, we get h = 3.

(c) The line ya divides R into two regions of equal area.

The area of R is 32/3 units².

Let the lower boundary of the **upper region **be y = a.

The corresponding x-coordinates are x = (a - 3)½ and x = (a - 3)½.So the area of the upper region is given by∫[(a - 3)½, 2] (4x + 3) dx - ∫[(a - 3)½, 2] (x² + 3) dx = 16/3 units²

We know that the area of the upper region is half the area of R. So,∫[(a - 3)½, 2] (4x + 3) dx - ∫[(a - 3)½, 2] (x² + 3) dx = 16/3 units²

We can solve this equation to get a = 13/3.

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If a 2-year capital project has an internal rate of return factor equal to 1.69 and net annual cash flows of $42000, the initial capital investment was

a. $24852.

b. $49704.

c. $70980.

d. $35490.

### Answers

The correct answer is option (a) $24,852.The initial **capital** investment for the 2-year capital project with an internal rate of return **factor** of 1.69 and net annual cash flows of $42,000 is $24,852.

The internal rate of return (IRR) is a measure used to evaluate the **profitability** of an **investment**. In this case, we know that the IRR factor is 1.69. The IRR factor is calculated by dividing the net present value (NPV) of the project by the initial capital investment. Since the IRR factor is given as 1.69, we can set up the equation: NPV / Initial **capital** investment = 1.69.

We also know that the net annual cash flows for the project are $42,000. The NPV can be calculated by multiplying the net annual cash flows by the IRR factor: NPV = Net annual cash flows × IRR factor. Plugging in the values, we get NPV = $42,000 × 1.69 = $70,980.

Now, we can rearrange the equation to solve for the initial capital investment: $70,980 / Initial capital investment = 1.69. **Cross-multiplying** and solving for the initial capital investment, we get: Initial capital investment = $70,980 / 1.69 = $24,852.

Therefore, the correct answer is option (a) $24,852.

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Consider a continuous-time system whose input x(t) and output y(t) are related by d² y(t) d1² + 2 dy(t) + y(t) = x(t) dt with initial conditions dy(0) = 1, y(0) = 1. dt (a) Determine the total response of the system for x(t) = sin t (b) Determine the total response as the sum of zero- input response and zero-state response for x(t) = sin t

### Answers

Therefore, the total response of the system can be found by adding the zero-input response and the** zero-state response**, which is given by:

y(t) = Y(s) * s^(-1) = [s^2 + 2s + 1] * s^(-1) = (s + 1) * e^(2s)

This is the total response of the system for x(t) = sin t and x(0) = 1.

To determine the total response of the system, we can use the transfer function H(s), defined as the ratio of the Laplace transforms of the output y(t) and input x(t), which is given by H(s) = Σ from n = -∞ to ∞ x(n) / Σ from n = -∞ to ∞ y(n)s^n.

The** Laplace transform **of the differential equation can be found as:

Y(s) = [s^2 + 2s + 1] * X(s)

Now, we can find the transfer function H(s) using the Laplace transforms of the initial conditions:

H(s) = [s^2 + 2s + 1] * [s^2 + 2s + 1]^-1 * [1, 1]^-1

Simplifying the denominator, we get:

H(s) = [s^2 + 2s + 1] * [s^2 + 2s + 1]^-1 * [1, 1]^-1 = [s + 1]^2

So, the transfer function of the system is H(s) = s + 1.

(b) To determine the total response as the sum of zero-input response and zero-state response, we need to find the Laplace transforms of the zero-input response and zero-state response of the system. The zero-input response of the system is the response of the system to a **constant input** of zero, which is x(t) = 0.

The zero-input response of the system can be found using the transfer function H(s), which is given by:

Y(s) = X(s) * H(s)

Substituting the Laplace transform of the input x(t) as 0, we get:

Y(s) = 0 * [s + 1]

So, the zero-input response of the system is Y(s) = 0.

The zero-state response of the system can be found using the transfer function H(s) and the Laplace transform of the** initial conditions**, which is given by:

Y(s) = X(s) * H(s) * Y(0)

Substituting the Laplace transform of the initial conditions as 1, we get:

Y(s) = X(s) * H(s) * Y(0) = [s + 1] * [s + 1] * 1 = [s^2 + 2s + 1]

So, the zero-state response of the system is Y(s) = [s^2 + 2s + 1].

Therefore, the **total response** of the system can be found by adding the zero-input response and the zero-state response, which is given by:

y(t) = Y(s) * s^(-1) = [s^2 + 2s + 1] * s^(-1) = (s + 1) * e^(2s)

This is the total response of the system for x(t) = sin t and x(0) = 1.

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let v = < -1, -2, 3 > and w = < 1, 1, 1 > and let l be the line through the point q = ( 1, 0, 1 ) in the direction of v the distance from the point q w to the line l is

### Answers

Given the** vectors** v = <-1, -2, 3> and w = <1, 1, 1>, and the point q = (1, 0, 1), we need to find the distance from the point q to the line l, which is defined as the line passing through q in the direction of v.

To find the distance from point q to line l, we can use the formula for the distance between a point and a line in** three-dimensional space**. The formula states that the distance d is equal to the length of the projection of the vector q - p onto the **normal vector** of the line, where p is a point on the line.

First, we can find a point p on the** line** l. Since the line passes through q in the direction of v, we can choose any scalar t and calculate p = q + tv. This will give us a point on the line l. Next, we can calculate the vector q - p. Subtracting the coordinates of p from the coordinates of q will give us the vector q - p.

Then, we can find the normal vector of the line l. Since v is in the direction of the line, it can serve as the normal vector. Alternatively, we can calculate the **cross product** of v and w to find the normal vector. Finally, we can find the length of the projection of the vector q - p onto the normal vector, which will give us the distance d.

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Suppose the rule 2\3 ƒ(−2,−1)+4ƒ (-2,0) + ƒ(−2,1)+ ƒ (2,−1)+4ƒ (2,0)+ƒ(2,1)] is applied to 12 solve [[ƒ(x,y) dx dy. Describe the form of the function ƒ(x, y) that are integrated -1-2 exactly by this rule and obtain the result of the integration by using this form.

### Answers

The given rule is applied to **integrate** the function ƒ(x, y) over a **rectangular region**. We need to determine the form of ƒ(x, y) that will yield an exact result using this rule.

To find the form of ƒ(x, y) that integrates exactly using the given rule, we examine the** coefficients** applied to the function values at each corner of the rectangular region. The coefficients are 2/3, 4, 1, and 1/3, corresponding to the corners (-2, -1), (-2, 0), (-2, 1), (2, -1), (2, 0), and (2, 1), respectively.

For ƒ(x, y) to integrate exactly, the function must be such that the sum of these coefficients multiplied by the function values at the respective corners equals the result of** integration**. This implies that ƒ(x, y) must satisfy specific relationships among its values at these corners.

By considering these conditions, we can determine the appropriate form of ƒ(x, y) that will yield an exact integration using the given rule. Once we have the** function **in this form, we can evaluate the integration by substituting the function values into the rule and calculating the result.

In summary, to integrate exactly using the given rule, the function ƒ(x, y) must satisfy specific conditions at the corners of the rectangular region. By examining the coefficients applied to the function **values**, we can determine the appropriate form of ƒ(x, y). Substituting the function values into the** rule** allows us to calculate the result of the integration.

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Krista Kellman has an opportunity to purchase a government security that will pay $200,000 in 5 years.

Note: Round answers to two decimal places.

1. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is 6% compounded annually.

2. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is 10% compounded annually.

3. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is 6% compounded semiannually.

### Answers

The price Krista would pay for the government security varies based on the **interest** rate and compounding frequency. For a 6% annual interest rate, the price is approximately $148,644.94. With a 10% annual interest rate, the price **decreases** to around $124,609.95. Lastly, if the interest is compounded semiannually at a rate of 6%, the price increases to approximately $149,758.84.

1. To calculate the price of the security with a 6% annual interest rate, we need to find the present value of the future $200,000 payment. Using the formula for present value of a single amount, we can calculate:

PV = FV / (1 + r)ⁿ

PV = 200,000 / (1 + 0.06)⁵

PV ≈ $148,644.94

Therefore, Krista would pay approximately $148,644.94 for the security if the interest rate is 6% compounded annually.

2. Similarly, if the interest rate is 10% compounded annually, we can calculate the **price** of the security:

PV = 200,000 / (1 + 0.10)⁵

PV ≈ $124,609.95

In this case, Krista would need to pay around $124,609.95 for the security.

3. If the **interest** rate is 6% **compounded** semiannually, we adjust the formula to account for the compounding frequency:

[tex]PV = 200,000 / (1 + (0.06/2))^{(5*2)}[/tex]

PV ≈ $149,758.84

Therefore, with a **semiannual** compounding frequency, Krista would pay approximately $149,758.84 for the security.

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The pulse rates of 168 randomly selected adult males vary from a low of 36 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want 95% confidence that the sample mean is within 4 bpm of the population mean. Complete parts (a) through (c) below a. Find the sample size using the range rule of thumb to estimate o. n= (Round up to the nearest whole number as needed.) b. Assume that o = 12.2 bpm, based on the value s = 12.2 bpm from the sample of 168 male pulse rates. n= (Round up to the nearest whole number as needed.) c. Compare the results from parts (a) and (b). Which result is likely to be better? The result from part (a) is the result from part (b). The result from is likely to be better because

### Answers

A) We need a minimum **sample size** of 119.

B) We need a minimum sample size of 89.

C) The result from part (a) is likely to be more accurate.

(a) To find the sample size using the range rule of thumb, we first need to calculate the range of the **pulse rates:**

range = highest value - lowest value = 104 bpm - 36 bpm = 68 bpm

Then, we can use the following formula to estimate the standard deviation:

s ≈ range / 4

Plugging in the values, we get:

s ≈ 68 / 4 = 17 bpm

To estimate the **minimum **sample size required to estimate the mean pulse rate of adult males with 95% confidence and a margin of error of 4 bpm, we can use the following formula:

n = (Zα/2)^2 * σ^2 / E^2

Where Zα/2 is the critical value for a 95% confidence level (which is 1.96), σ is the estimated standard deviation (which is 17 bpm using the range rule of thumb), and E is the desired **margin** of error (which is 4 bpm).

Plugging in the values, we get:

n = (1.96)^2 * 17^2 / 4^2 ≈ 119

Therefore, we need a minimum sample size of 119.

(b) Using the given** value** of s = 12.2 bpm from the sample of 168 male pulse rates, we can directly substitute it into the formula for calculating the minimum sample size:

n = (Zα/2)^2 * σ^2 / E^2

n = (1.96)^2 * 12.2^2 / 4^2 ≈ 89

Therefore, we need a minimum sample size **of 89.**

(c) The result from part (a) is likely to be better because it is based on a larger estimated standard deviation. When the estimated standard deviation is **smaller, **as in part (b), the sample size required to achieve the desired level of precision is larger. In other words, a larger sample size is needed to achieve the same level of precision when the estimated standard deviation is smaller. Therefore, the result from part (a) is likely to be more accurate.

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find the equation of the line tangent to the graph of f at the indicated x value. y=5sin⁻¹5x, x=0

Tangent line: y =

### Answers

The equation of the **tangent **line to the graph of f at x = 0 is y = 25x.

The equation of the tangent line to the graph of f at the indicated x value can be found by taking the **derivative** of the function f and evaluating it at the given x value.

The derivative of y = 5sin⁻¹(5x) can be obtained using the chain rule. Let's denote the derivative as f'(x). Then, the equation of the tangent line is given by y = f'(x)(x - x₀) + f(x₀), where x₀ is the given x value.

To find the derivative, we **differentiate **the function y = 5sin⁻¹(5x) with respect to x. Using the chain rule, we have:

f'(x) = d/dx [5sin⁻¹(5x)]

= 5 * d/dx [sin⁻¹(5x)]

= 5 * (1/√(1 - (5x)²)) * d/dx [5x]

= 5 * (1/√(1 - 25x²)) * 5

Simplifying this **expression**, we have:

f'(x) = 25/√(1 - 25x²)

Now, substituting x = 0 into the equation, we have x₀ = 0:

f'(0) = 25/√(1 - 25(0)²)

= 25

Therefore, the equation of the tangent **line **is given by y = 25(x - 0) + 5sin⁻¹(5(0)), which simplifies to y = 25x.

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1 Consider the functions f(x) = − 2x + 9 and g(x) = -(x − 9). (a) Find f(g(x)). (b) Find g(f(x)). (c) Determine whether the functions f and g are inverses of each other.

### Answers

(a) f(g(x)) = 2x - 9

(b) g(f(x)) = -2x + 18

(c) Yes, the functions f and g are **inverses **of each other.

How to find f(g(x)), we substitute g(x) into the function f(x)?

(a) To find f(g(x)), we substitute g(x) into the **function **f(x). We have g(x) = -(x - 9), so:

f(g(x)) = -2(g(x)) + 9 = -2(-(x - 9)) + 9

Simplifying this expression, we distribute the negative sign:

f(g(x)) = -2(-x + 9) + 9

Multiplying -2 by each term inside the **parentheses **gives:

f(g(x)) = 2x - 18 + 9

Combining like terms, we get:

f(g(x)) = 2x - 9

How to find g(f(x))?

(b) Similarly, to find g(f(x)), we substitute f(x) into the function g(x). We have f(x) = -2x + 9, so:

g(f(x)) = -(f(x) - 9) = -(-(2x - 9) - 9)

Again, we distribute the negative sign:

g(f(x)) = -(2x - 9 + 9) = -(2x - 18)

Simplifying, we get:

g(f(x)) = -2x + 18

How to find whether the functions f and g are inverses of each other?

(c) To determine whether the functions f and g are **inverses **of each other, we need to check if f(g(x)) = x and g(f(x)) = x.

From part (a), we found that f(g(x)) = 2x - 9. To check if f(g(x)) = x, we set 2x - 9 equal to x and solve for x:

2x - 9 = x

x = 9

Therefore, f(g(x)) = x for x = 9.

From part (b), we found that g(f(x)) = -2x + 18. To check if g(f(x)) = x, we set -2x + 18 equal to x and solve for x:

-2x + 18 = x

3x = 18

x = 6

Therefore, g(f(x)) = x for x = 6.

Since f(g(x)) = x and g(f(x)) = x for specific values of x, we can conclude that the functions f and g are inverses of each other.

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Leonhard Euler was able to calculate the exact sum of the p-series with p = 2: [infinity]Σₙ₌₁ 1/n² = π²/6 Use this fact to find the sum of each series. [infinity]Σₙ₌₅ 1/n² = [infinity]Σₙ₌₂ 1/(n+3)² = [infinity]Σₙ₌₁ 1/(2n)² = [infinity]Σₙ₌₀ 1/(2n+1)² =

### Answers

**Leonhard Euler's** calculation of the exact sum of the p-series with p = 2, which is given by [infinity]Σₙ₌₁ 1/n² = π²/6, can be used to find the sum of several related series.

By manipulating the given series, we can express them in a form that matches Euler's result and use the known value of π²/6 to find their sums. The sums of the series [infinity]Σₙ₌₅ 1/n², [infinity]Σₙ₌₂ 1/(n+3)², [infinity]Σₙ₌₁ 1/(2n)², and [infinity]Σₙ₌₀ 1/(2n+1)² can be determined based on this fact.

Euler's result states that the sum of the series** [infinity]Σₙ₌₁ 1/n² **is equal to **π²/6. **We can use this result to find the sums of other related series.

[infinity]Σₙ₌₅ 1/n²:

We can rewrite this series as [infinity]Σₙ₌₁ 1/n² - [infinity]Σₙ₌₁ 1/k², where k **ranges** from 1 to 4. By applying Euler's result, the sum of this** series** is equal to π²/6 - 1²/1² - 1²/2² - 1²/3² - 1²/4².

[infinity]Σₙ₌₂ 1/(n+3)²:

We can rewrite this series as [infinity]Σₙ₌₁ 1/n² by substituting n+3 with n. Therefore, the sum of this series is equal to π²/6.

[infinity]Σₙ₌₁ 1/(2n)²:

We can rewrite this series as [infinity]Σₙ₌₀ 1/n² by substituting 2n with n. Therefore, the sum of this series is equal to π²/6.

[infinity]Σₙ₌₀ 1/(2n+1)²:

We can rewrite this series as [infinity]Σₙ₌₁ 1/n² - [infinity]Σₙ₌₁ 1/k², where k ranges from 2 to ∞. By applying Euler's result, the sum of this series is equal to π²/6 - 1²/2² - 1²/3² - 1²/4² - ... - 1²/k².

In each case, the sum of the series can be found by using Euler's result and manipulating the series to match the given form.

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Which equation below describes the line passing through the points (3 , -2) and (5 , 4)?

### Answers

The equation of a line is y=mx + b

m is the slope and b is the y-intercept (where on the y axis the line intersects it)

We can use the formula (y2-y1) divided by (x2-x1) to find the slope, m

Therefore (4+2)/(5-3)

This is 6/2 which is 3

The equation is now y=3x + b

Now we use one point and plug into the equation to solve for b

4=3(5) + b

4=15 + b

Therefore b is -11

The equation is y=3x-11

Graph the trigonometric function.

Y=3/2 sin2x

Plot all points corresponding to x-intercepts, minima, and maxima within one cycle. Then click on the graph-a-function button.

### Answers

The **x-intercepts** is x = 0 and x = π.

The **Minima and maxima** is x = π/2 and x = 3π/2.

To graph the **trigonometric** **function** y = (3/2)sin^2(x) and plot the x-intercepts, minima, and maxima within one cycle, we'll first identify these key points.

The general form of the **equation** y = (3/2)sin^2(x) is y = (3/2)(sin(x))^2, where sin(x) represents the standard sine function.

Key points to plot within one cycle (from 0 to 2π):

1. x-intercepts: These occur when sin(x) = 0. So, x = 0 and x = π.

2. Minima and maxima: These occur when sin(x) takes its **extreme** values of -1 and 1. So, x = π/2 and x = 3π/2.

Using these points, we can plot the graph:

**Plotting** the x-intercept at x = 0: (0, 0)

Plotting the x-intercept at x = π: (π, 0)

Plotting the minimum at x = π/2: (π/2, 3/2)

Plotting the maximum at x = 3π/2: (3π/2, 3/2)

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If the r v. X is distributed as uniform distribution over [-beta. Bata]. where beta > 0, Determine the parameter beta, so that each of the following equalities holds: a. P(-1 < X < 1) = 0.75. b. P(|X| > 2) = 0.5

### Answers

(a) In a** uniform distribution**, the probability density function (PDF) is given by f(x) = 1 / (beta - alpha) for x in the interval [alpha, beta]. In this case, we have **alpha **= -beta and we need to find the value of beta such that P(-1 < X < 1) equals 0.75.

Since the PDF is constant within the interval, we can calculate this probability as the ratio of the **interval length **(-1 to 1) to the total length (2 * beta). Therefore, 2 / (2 * beta) = 0.75, which **simplifies **to beta = 2 / 0.75.

(b) To find the value of beta for which P(|X| > 2) equals 0.5, we consider that P(|X| > 2) is equivalent to 1 - P(-2 < X < 2). Using the same approach as in part (a), we calculate the probability as (2 * 2) / (2 * beta) = 0.5, which simplifies to beta = 2 * 2 / 0.5.

By solving the **equations **in both parts (a) and (b), we can find the value of beta that satisfies the given **conditions **for the uniform distribution over the interval [-beta, beta].

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