Please Help. All I have been getting is bot answers, and I actually need help with this. Thanks.



1. Two neighbors are each hosting a party. The first neighbor orders 5 large pizzas, each with a diameter of 16 inches. The second neighbor orders 9 small pizzas, each with a diameter of 12 inches. In terms of area, which party has more pizza? Explain. Show all work.



2. A toy train set has a circular track piece. The inner radius of the piece is 6 cm. One sector of the track has an arc length of 33 cm on the inside and 55 cm on the outside. What is the width of the track?

The correct statement that compares the value of the 6 in 26.495 and 17.64 is the value of the 6 in 26.495 is 10 times the value of the 6 in 17.64. Therefore, the correct option is D.

This is because the value of a digit is determined by its place in the number. In 26.495, the 6 is in the tenths place, which means it represents 6/10 or 0.6. In 17.64, the 6 is in the hundredths place, which means it represents 6/100 or 0.06. Therefore, the value of the 6 in 26.495 is 0.6 and the value of the 6 in 17.64 is 0.06.

To compare these values, we can divide the value of the 6 in 26.495 by the value of the 6 in 17.64. This gives us 0.6/0.06 = 10. Therefore, the value of the 6 in 26.495 is 10 times greater than the value of the 6 in 17.64 which corresponds to option D.

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The pair of numbers  3/5 and -5/3,  -1/3 and 3 are perpendicular because two lines are perpendicular if and only if the product of their gradients is -1.

Each pair of numbers, we have to decide if lines with these gradients are perpendicular or not

Two lines are perpendicular if and only if the product of their gradients is -1.

For 5 and 1/5

The product is 1 which is not -1, so these are not perpendicular.

For 3/5 and -5/3

The product is -1 so these are perpendicular

For 1/4 and -1/4

The product is -1/16 so these are not perpendicular.

For -1/3 and 3

The product is  -1 so these are perpendicular

Hence, the pair of numbers  3/5 and -5/3,  -1/3 and 3 are perpendicular.

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The substitution (3x - 2)2 u = 3x - 2 transforms the indefinite integral da into none of the given options. It should result in the integral of the function being expressed in terms of u rather than x.

This substitution is an example of using a change of variables to simplify an integral by transforming it into a more manageable form. This can be particularly useful when dealing with complicated integrals that are difficult to solve by other methods. Additionally, using such transforms can often provide insight into the underlying structure of the problem being studied.
 Based on your question, it appears that you want to perform a substitution to transform the indefinite integral of "da" using the substitution (3x - 2)² u = 3x - 2. However, the given integral "da" doesn't seem to be correct or complete. Please provide the complete integral, and I will be happy to help you with the transformation using the given substitution.

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Every day, a baker produces 20 loaves of bread. The loaves come in white or brown. It would take him 30 days to make 180 loaves of brown bread.

Let the ratio be x,

We have been given the ratio of 7 : 3 in which 3 part is of brown bed.

So the number of brown bread will be 3x and white bread will be 7x.

Now, we have to find out the days for 180 loaves of brown bread. This means that 3x = 180. Now we will find out the value of x to find the total loaves of bread from this equation.

3x = 180

x = 180 / 3

x = 60

So, the total amount of bread = brown bread + white bread

Total amount of bread = 3x + 7x

= 10x

= 10 × 60

= 600

We know that in 20 loaves of bread are made in one day, so

time taken to make 600 loaves of bread = 600 / 20

= 30 days

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A triangular prism is a three-dimensional shape with two parallel triangular bases and three rectangular faces. In this case, the triangular bases have sides of 4 cm, 5 cm, and 5 cm, while the rectangular faces have a length of 10 cm and a height of 6 cm.

To find the volume of the prism, we can use the formula V = Bh, where B is the area of the base and h is the height. The area of a triangle can be found using the formula A = 1/2bh, where b is the base and h is the height.

So, for the triangular base of this prism, we have:

A = 1/2(4 cm)(3 cm) = 6 cm²

The height of the prism is 5 cm, so:

V = Bh = (6 cm²)(5 cm) = 30 cm³

Therefore, the volume of the prism is 30 cubic centimeters.

To find the surface area of the prism, we need to calculate the area of each face and add them up.

The two triangular faces each have an area of:

A = 1/2(4 cm)(5 cm) = 10 cm²

And the three rectangular faces each have an area of:

A = (10 cm)(6 cm) = 60 cm²

So, the total surface area is:

SA = 2(10 cm²) + 3(60 cm²) = 200 cm²

Therefore, the surface area of the prism is 200 square centimeters.

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If that segment KL is parallel to segment MN and that segment KN bisects segment ML, then segment KO is congruent to segment NO.

To prove that segment KO is congruent to segment NO, we need to show that triangle KNO is an isosceles triangle, with KO ≅ NO.

From the given information, we know that KL is parallel to MN, which means that angle KLN is congruent to angle MNL (corresponding angles). Also, KN bisects segment ML, which means that angle KNO is congruent to angle NMO (angle bisector theorem).

Therefore, we have:

angle KNO = angle NMO

angle KLN = angle MNL

Adding these two equations gives us:

angle KNO + angle KLN = angle NMO + angle MNL

But angle KLN + angle NMO + angle MNL = 180 degrees (as they form a straight line). So we can substitute this into the equation:

angle KNO + 180 degrees = 180 degrees

Simplifying, we get:

angle KNO = 0 degrees

This means that KO and NO are on the same line, so they must be congruent. Therefore, we have proven that segment KO is congruent to segment NO.

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Answer:

1.03x 10^{10}

Step-by-step explanation:

No explanation, simple calculator calculation does the job.

Your answer will depend on the measurements you obtain from the parallelograms.

To determine the scale factor of the dilation between the green parallelogram and the black parallelogram, follow these steps:
1. Choose corresponding sides of both parallelograms (e.g., the base or the height).
2. Measure the length of the chosen side in the green parallelogram and the same side in the black parallelogram.
3. Divide the length of the side in the green parallelogram by the length of the corresponding side in the black parallelogram.
The result will be the scale factor of the dilation. Compare the result with the given options:
A) 1/3
B) 1/2
C) 2
Your answer will depend on the measurements you obtain from the parallelograms.

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The cordinate points with x- and y-intercepts of the graph of a linear equation, 4x+ 8y = 20, are equals to the (5,0) and (0, 5/2).

We have an equation, 4x + 8y = 20 --(1) which is linear equation with two variables. We have to determine the the x- and y-intercepts of the graph of equation (1). The graph of line (1) is present in above figure. Slope intercept form of equation (1) is written as [tex]y = - \frac{1}{2}x + \frac{5}{2}[/tex],

The x-intercept is point where a line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. As we know, two points determine any line, we can graph lines using the x- and y-intercepts. To determine the x-intercept, we substitute y=0 and solve for x. So, when y = 0 then 4x + 0 = 20

=> x = 5

similarly to determine the y-intercept, set x=0 and solve for y. When x = 0

=> 8y = 20

=> y = 5/2.

Hence, required value are (5,0) and (0,5/2).

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Hi! I'd be happy to help you with your question about transformations and congruence. When Triangle ABC is reflected over the x-axis, the algebraic rule applied to the figure is:

Your answer: (x, y) → (x, -y)

This rule states that the x-coordinate remains the same, while the y-coordinate is multiplied by -1, resulting in a reflection over the x-axis. This transformation preserves congruence, as the size and shape of Triangle ABC remain the same, only its position changes.

The algebraic rule applied to the figure when Triangle ABC is reflected over the x-axis is (x,y) → (x,-y), where x represents the x-coordinate and y represents the y-coordinate. This is because reflecting a figure over the x-axis involves keeping the x-coordinate the same while changing the sign of the y-coordinate. This preserves the congruence of the original and reflected triangles.

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At the end of 5 years, Sam will have paid $1,900 in interest to the bank. This calculation assumes that the interest rate remains constant and is applied on a simple basis, rather than being compounded over the 5-year period.

Sam makes a cash deposit of $8,000 on a used car and is charged an interest rate of 4.75% per year by the bank. To calculate the interest he will pay at the end of 5 years, we can use the formula for simple interest, which is I = P × R × T, where I is the interest, P is the principal (the initial deposit), R is the interest rate, and T is the time in years.

In this case, P = $8,000, R = 4.75% (or 0.0475 in decimal form), and T = 5 years. Plugging these values into the formula, we get:

I = $8,000 × 0.0475 × 5

I = $1,900

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Sam will pay $1,900 in interest to the bank at the end of 5 years.

The interest rate is 4.75% per year, which can be written as 0.0475 as a decimal. The amount of interest that Sam will pay after 5 years can be calculated using the simple interest formula:

Interest = Principal x Rate x Time

where Principal is the initial deposit, Rate is the interest rate, and Time is the length of time the interest is charged for.

In this case, the Principal is $8,000, the Rate is 0.0475, and the Time is 5 years. Plugging these values into the formula, we get:

Interest = $8,000 x 0.0475 x 5

Interest = $1,900

Therefore, Sam will pay $1,900 in interest to the bank at the end of 5 years.

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The cost of a croissant before the increase was $0.75, the cost of a cup of coffee was $0.75, and the cost of a fruit bowl was $2.25.

Let's start by assigning variables to the cost of each item before the price increase. Let x be the cost of a croissant, y be the cost of a cup of coffee, and z be the cost of a fruit bowl.

From the problem statement, we know that:

x + y + z = 5.25 (total cost before price increase)

z = 3x (fruit bowl costs 3 times as much as a croissant)

Substituting z = 3x into the first equation, we get:

x + y + 3x = 5.25

4x + y = 5.25

Now we need to solve for x and y. We don't have an equation directly relating the price increase to the new prices, but we can use the percentage increase to write:

New croissant price = x + 0.15x = 1.15x

New coffee price = y + 0.4y = 1.4y

The new total cost will be:

1.15x + 1.4y + z

Substituting z = 3x, we get:

1.15x + 1.4y + 3x

Simplifying this expression and using the equation 4x + y = 5.25 to eliminate y, we get:

1.15x + 1.4y + 3x = 4.15x + 1.4(5.25 - 4x)

4.15x + 1.4(4x - 5.25) = 4.55x - 5.85

Therefore, the new total cost will be $4.55x - $5.85. To find the cost of each item before the increase, we can solve the system of equations:

4x + y = 5.25

z = 3x

Substituting z = 3x into the first equation, we get:

4x + y + 3x = 5.25

7x + y = 5.25

Solving for y in terms of x, we get:

y = 5.25 - 7x

Substituting this expression into the equation for the new total cost, we get:

4.55x - 5.85 = 1.15x + 1.4(5.25 - 4x) + 3x

Simplifying and solving for x, we get:

x = 0.75

Substituting this value of x into the equation for y, we get:

y = 5.25 - 7(0.75) = 0.75

Substituting x and z = 3x into the equation for the total cost before the increase, we get:

0.75 + 0.75 + 3(0.75) = 3.75

Therefore, the cost of a croissant before the increase was $0.75, the cost of a cup of coffee was $0.75, and the cost of a fruit bowl was $2.25.

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You need to pay $3,243.50 in property taxes.

The appraised value of the house after setting out the homestead exemption is $315,000 - $50,000 = $265,000.

To calculate the assets taxes, we need to multiply the appraised cost by means of the tax rate, which is 1.22% or 0.0122 as a decimal:

property taxes = $265,000 x zero.0122 = $3,243.50

Therefore, you have to pay $3,243.50 in property taxes.

It's far essential to factor in property taxes whilst thinking about the general price of purchasing a home and to recognize the method for applying for exemptions or appealing the appraised cost if necessary.

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The annual rate as a percent is approximately 4.86% per year.

To find the annual rate as a percent, we first need to understand the function that Robert is using. The function [tex]f(x) = 1000(1.0375)^x[/tex] represents the amount of money Robert earns each year based on his initial investment of $1000 and the interest rate of 3.75% (as represented by the value 1.0375, which is 1 + 0.0375).

To calculate the annual rate as a percent, we need to isolate the interest rate from the function. We can do this by using logarithms. Taking the logarithm of both sides of the equation, we get:

log(f(x)) = log(1000) + x*log(1.0375)

Now, we can solve for the interest rate (represented by the value of 1.0375) by dividing both sides of the equation by x and then taking the antilog of the result:

1.0375 = antilog[(log(f(x)) - log(1000))/x]

Using this formula, we can plug in any value for x (representing the number of years Robert has held his investment) and find the corresponding interest rate. For example, if Robert has held his investment for 5 years, we can calculate the interest rate as:

1.0375 = antilog[(log(f(5)) - log(1000))/5]

This gives us an interest rate of approximately 4.86% per year. So, to answer the original question, the annual rate as a percent is approximately 4.86%.

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Answer:

The function g(x) = (3x)² can be simplified to g(x) = 9x², which is a vertical stretch of f(x) = x² by a factor of 9.

Therefore, the correct answer is B. The graph of g(x) is vertically stretched by a factor of 3 compared to the graph of f(x).

The surface area of revolution of C about the x-axis is π/27 (81^(3/2) - 1) or approximately 478.48 units².

To find the surface area of revolution of C about the x-axis, we can use the formula:

Surface area = ∫2πy ds

where y is the function that defines the curve C, and ds is an element of arc length along the curve.

We can express ds in terms of dx as follows:

ds = √(1 + (dy/dx)²) dx

where dy/dx is the derivative of y with respect to x.

For the curve C, we have:

y = 3x³

dy/dx = 9x²

Substituting these into the expression for ds, we get:

ds = √(1 + (9x²)²) dx

= √(1 + 81x⁴) dx

Substituting y and ds into the formula for surface area, we get:

Surface area = ∫₂πy √(1 + (dy/dx)²) dx

= ∫₀³ 2π(3x³) √(1 + 81x⁴) dx

This integral can be evaluated using substitution:

Let u = 1 + 81x⁴

Then du/dx = 324x³

And dx = du/324x³

Substituting these into the integral, we get:

Surface area = ∫₁₀³ 2π(3x³) √(1 + 81x⁴) dx

= 2π/108 ∫₁₀³ (3x³) √u du

= π/54 ∫₁₀³ u^(1/2) du

= π/54 (2/3) u^(3/2) | from 1 to 81

= π/81 (2/3)(81^(3/2) - 1)

= π/27 (81^(3/2) - 1)

Therefore, To find the surface area of revolution of C about the x-axis, we can use the formula:

Surface area = ∫2πy ds

where y is the function that defines the curve C, and ds is an element of arc length along the curve.

We can express ds in terms of dx as follows:

ds = √(1 + (dy/dx)²) dx

where dy/dx is the derivative of y with respect to x.

For the curve C, we have:

y = 3x³

dy/dx = 9x²

Substituting these into the expression for ds, we get:

ds = √(1 + (9x²)²) dx

= √(1 + 81x⁴) dx

Substituting y and ds into the formula for surface area, we get:

Surface area = ∫₂πy √(1 + (dy/dx)²) dx

= ∫₀³ 2π(3x³) √(1 + 81x⁴) dx

This integral can be evaluated using substitution:

Let u = 1 + 81x⁴

Then du/dx = 324x³

And dx = du/324x³

Substituting these into the integral, we get:

Surface area = ∫₁₀³ 2π(3x³) √(1 + 81x⁴) dx

= 2π/108 ∫₁₀³ (3x³) √u du

= π/54 ∫₁₀³ u^(1/2) du

= π/54 (2/3) u^(3/2) | from 1 to 81

= π/81 (2/3)(81^(3/2) - 1)

= π/27 (81^(3/2) - 1)

Therefore, the surface area of revolution of C about the x-axis is π/27 (81^(3/2) - 1) or approximately 478.48 units².

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The length of the trail is determined as 1.3 miles.

The length of the trail is calculated as follows;

The circumference of the circle is calculated as;

S = πd

where;

S = π x 2 ft

S = 2π ft

I revolution = 1 circumference = 2π ft

1 rev = 2π ft

1,100.5 rev = ?

= 1,100.5 rev/rev x 2π ft

= 6,914.65 ft

5280 ft -------> 1 mile

6,914.65 ft ------> ?

= 1.3 miles

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The formula for d is d = (t * 21/4b² + 3b)/5

To make d the subject of the formula t=4b²/21(d-3b/5), we need to isolate d on one side of the equation and simplify.

First, let's simplify the right side of the equation by multiplying the fraction by the LCD of 5:

t = 4b²/21(d-3b/5)

t = (4b²/21d) * 5d - 3b

Now, we can isolate d by dividing both sides of the equation by the coefficient of d on the right side:

t/(4b²/21) = 5d - 3b

Simplifying the left side, we get:

t * 21/4b² = 5d - 3b

Adding 3b to both sides of the equation, we get:

t * 21/4b² + 3b = 5d

Finally, we can divide both sides by 5 to isolate d:

d = (t * 21/4b² + 3b)/5

Therefore, the formula for d is:

d = (t * 21/4b² + 3b)/5

In words, to find the value of d, we need to multiply the value of t by 21/4b², add 3b to the result, and divide the sum by 5.

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The set of dimensions for the largest box is: 4 in x 4 in x 3.8 in.

We know that the smallest box must hold 8 fluid ounces or 15 in³ of soup. Let's assume the dimensions of the smallest box to be x, y, and z.

Then, we have:

[tex]x * y * z = 15[/tex]

Now, the largest box will be a dilation of the smallest box using a scale factor of 2. This means that every dimension of the smallest box will be multiplied by 2 to get the dimensions of the largest box.

So, the dimensions of the largest box will be 2x, 2y, and 2z.

Now, we need to find the dimensions of the smallest box. We can start by solving the equation x * y * z = 15 for one of the variables, say z:

[tex]z = 15 / (x * y)[/tex]

Substituting this value of z in the expression for the dimensions of the largest box, we get:

[tex]2x * 2y * (15 / (x * y))[/tex]

Simplifying this expression, we get:

[tex]4 * 15 = 60[/tex]

So, the dimensions of the largest box are approximately 4 in by 4 in by 3.8 in (rounded to the nearest tenth).

Therefore, the set of dimensions for the largest box is: 4 in x 4 in x 3.8 in.

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Each game of skee ball requires 2 tokens and each game of pinball requires 2 tokens.

Let the number of tokens required for each game of skee ball be x and for each game of pinball be y.

From the given information, we can form two equations:

10x + 8y = 44    ... (1)

3x + 8y = 30      ... (2)

Multiplying equation (2) by 3, we get:

9x + 24y = 90     ... (3)

Subtracting equation (1) from equation (3), we get:

- x + 16y = 46

Solving for x, we get:

x = 16y - 46

Substituting this value of x in equation (2), we get:

3(16y - 46) + 8y = 30

Simplifying and solving for y, we get:

y = 2

Substituting this value of y in equation (1), we get:

10x + 8(2) = 44

Solving for x, we get:

x = 2

Therefore, each game of skee ball requires 2 tokens and each game of pinball requires 2 tokens.

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The angle that the lengths of ropes make with the ceiling is 90 degrees.

To determine the angle that the lengths of ropes make with the ceiling for an object with a weight of 100 N suspended by two ropes with equal tension of 50 N, we can follow these steps:

1. Understand that the vertical forces must balance, meaning the sum of the vertical components of tension in each rope must equal the object's weight.

2. Recognize that the vertical component of tension in each rope can be calculated using the sine function and the angle, θ, between the rope and the ceiling: T_vertical = T * sin(θ).

3. Set up an equation using the information provided: 2 * (50 N * sin(θ)) = 100 N, where θ is the angle we want to find.

4. Simplify the equation: 100 * sin(θ) = 100 N.

5. Divide both sides by 100: sin(θ) = 1.

6. Find the inverse sine (also known as arcsin) of 1: θ = arcsin(1).

7. Calculate the angle: θ = 90 degrees.

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Therefore, the correct answer is option (b) c + 0.92.

The expression that represents Evelyn's grocery bill after the 8% discount is:

c - 0.08c

This can be simplified as:

0.92c

Therefore, the correct answer is option (b) c + 0.92.

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The expression that represents Evelyn's grocery bill after the coupon is applied is 0.92c. Therefore, the correct option is D.

If Evelyn has a coupon that will reduce her grocery bill by 8% and c represents the cost of her groceries, the expression that represents Evelyn's grocery bill after using the coupon is 0.92c. It is determined as follows.

1. The coupon reduces the bill by 8%, which means Evelyn will pay 100% - 8% = 92% of the original cost.

2. Convert the percentage to a decimal: 92% = 0.92

3. Multiply the original cost (c) by the decimal: 0.92c

So, the correct answer is option D: 0.92c.

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The number of times of rolling a 5 or a 6 in the fair die is 24

From the question, we have the following parameters that can be used in our computation:

Fair 6-sided die = 72 times

In a 6-sided die, we have

P(5 or 6) = 2/6

When evaluated, we have

P(5 or 6) = 1/3

So, when the die is rolled 72 times, we have

Expected value = 1/3 * 72

Evaluate

Expected value = 24

Hence, the number of times is 24

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Ramiro is correct that if all corresponding angles are equal then the triangle is said to be similar.

Triangles are said to be similar if any of the following is true:

1. All or any two of the corresponding angles are equal

2. All the corresponding sides are proportional to each other

3. One of the corresponding angles is equal and the adjoining corresponding sides are proportional.

Triangles are said to be congruent if any of the following is true:

1. All of the corresponding sides are equal

2. Two of the angles are equal and so is one of the corresponding sides of the triangle.

3. One of the corresponding angles is equal and the adjoining corresponding sides are also equal.

4. In a right-angled triangle, either the base or height and the hypotenuse are equal.

Since in the question, the criteria for the similar triangles is fulfilled then Ramiro is correct.

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The dimensions that require the least amount of material are r = (10/π)^(1/3) cm for the radius and h = (10/π)^(1/3) cm for the height.

To minimize the amount of material used for making aluminum cups with a volume of 10 cubic centimeters, you will need to optimize the dimensions of the right circular cylinders.

Given the volume (V) is 10 cm³, the formula for the volume of a right circular cylinder is V = πr²h, where r is the radius and h is the height.

10 = πr²h

To minimize the material used, we want to minimize the surface area (SA) of the open cylinder, which is given by the formula SA = 2πrh + πr² (the first term represents the lateral surface and the second term the base).

Using the volume formula, we can find a relationship between r and h:

h = 10 / (πr²)

Now substitute this expression for h in the surface area formula:

SA(r) = 2πr(10 / (πr²)) + πr²
SA(r) = 20/r + πr²

To find the minimum surface area, differentiate SA(r) with respect to r and set the result equal to zero:

d(SA)/dr = -20/r² + 2πr

Now solve for r:

0 = -20/r² + 2πr
20/r² = 2πr
r³ = 10/π

Now take the cube root of both sides:

r = (10/π)^(1/3)

To find the height, substitute this value of r back into the expression for h:

h = 10 / (π((10/π)^(1/3))²)
h = (10/π)^(1/3)

The dimensions that require the least amount of material are r = (10/π)^(1/3) cm for the radius and h = (10/π)^(1/3) cm for the height.

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Answer:

Since it is a right triangle, we can apply pythagores theorem.

Answer: a = 8.7 miles

Step-by-step explanation:

a^2 = c^2 - b^2

a^2 = 10^2 - 5^2

a^2 = 100 - 25

a^2 = 75

a ≈ 8.7

Therefore, the length of the missing leg is approximately 8.7 miles.

The probability of drawing a yellow marble on the next draw is 13/48, which is option A, 13%.

The probability of drawing a yellow marble in the next draw depends on whether the drawing process is with or without replacement.

If the drawing process is with replacement, meaning that the marble is put back into the bag after each draw, then the probability of drawing a yellow marble remains the same at 13/48.

If the drawing process is without replacement, meaning that the marble is not put back into the bag after each draw, then the probability of drawing a yellow marble changes. After 48 draws, there are 35 white marbles and 13 yellow marbles left in the bag.

Therefore, the correct answer is A) 13%.

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Answer: is approximately between 113.39 and 116.61
To calculate the 95% confidence interval for the students' average IQ score, we'll use the given information: sample size (n=150), sample mean (X=115), and sample standard deviation (s=10). We'll use the t-distribution since the population standard deviation is unknown.

First, we need to find the t-value for a 95% confidence interval with n-1 (149) degrees of freedom. Using a t-table or calculator, we find the t-value to be approximately 1.976.

Next, we'll calculate the standard error (SE) using the formula: SE = s/√n. In this case, SE = 10/√150 ≈ 0.816.

Now, we can find the margin of error (ME) using the formula: ME = t-value × SE. For this problem, ME = 1.976 × 0.816 ≈ 1.61.

Finally, to calculate the 95% confidence interval, we'll use the formula: X ± ME. Thus, the 95% confidence interval is 115 ± 1.61, which is approximately (113.39, 116.61).

So, the 95% confidence interval for the students' average IQ score is approximately between 113.39 and 116.61.

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(d) Probability of at least one major earthquake in the next 49 years is 0.9884, and (e) the probability of no major earthquakes in the next 49 years is 0.0116.

We will use the Poisson distribution to compute the probabilities. First, we need to find the value of λ (average number of earthquakes in a given time period).

(d) Compute the probability of at least one major earthquake in the next 49 years:

1. Calculate λ for 49 years:

(2.0 earthquakes / 22 years) * 49 years = 4.45 (rounded to the nearest hundredth)

2. Compute the probability of no major earthquakes (P(0)) in the next 49 years using Poisson distribution formula:

P(0) = (e^(-λ) * (λ^0)) / 0! = (e^(-4.45) * (4.45^0)) / 1 = 0.0116 (rounded to 4 decimal places)

3. Compute the probability of at least one major earthquake:

P(at least 1) = 1 - P(0) = 1 - 0.0116 = 0.9884

So, the probability of at least one major earthquake in the next 49 years is 0.9884 (rounded to 4 decimal places).

(e) Compute the probability of no major earthquakes in the next 49 years:

As calculated in step (d), the probability of no major earthquakes in the next 49 years is 0.0116 (rounded to 4 decimal places).

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Therefore, a person starts with a hand of cards without any Wild (Wild or Wild draw four) is 0.5741

and a number of outcomes of event 'B' is 16007560800.

total number of cards = N = 108

wild cards = 4

wild draw four = 4

we have 4 different colors and each color has 25 cards with break down as

zero numbered card = 1

numbered from 1 to 9 = 2 each (total 18)

skip = 2

draw two = 2

reverse = 2

skip, draw two and reverse are action cards

therefore, in a deck of UNO we have 3*2*4 = 24 action cards

a seven-card hand is dealt to each player from a well shuffled pack of card.

therefore, as each card is equally likely, we have total possible outcomes as 108C7

Ω is a event of every possible outcome and it has 108C7 = 27883218168 elements

an equally likely model is a model where equal weights are attached to every outcome and thus the probability of each outcomes become equal.

1) let 'A' be the event a whole hand is of action cards.

we have a total of 24 action cards in a UNO deck of 108 cards.and we have to draw a hand of 7 cards

total number of outcomes for event 'A' is 24C7

therefore, the probability that a player starting with a whole hand of action cards is

= [tex]24C7\n108C7\n[/tex]

= 0.0000124

therefore, a person starts with a whole hand of action card is 0.0000124

and the number of outcomes of event 'A' is 346104

2) let 'B' be the event that a hand of card is without any wild card

therefore we have a total of 100 cards that do not have any wild ( wild or wild draw four) cards

therefore a hand of 7 out of 100 cards for event 'B' can be drawn in 100C7 total ways

therefore, the probability that a player starts with a hand of cards without any Wild (Wild or Wild draw four) cards is

= [tex]100C7\n108C7\n[/tex]

= 0.5741

therefore, a person starts with a hand of cards without any Wild (Wild or Wild draw four) is 0.5741

and the number of outcomes of event 'B' is 16007560800.

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