Does the transformation appear to be a rigid motion?

The values of U and d for an arithmetic sequence with U20 = 100 and U25 = 115 is U =  43 and d = 3.

The formula for the nth term of an arithmetic sequence: Un = U1 + (n-1)d

We know that U20 = 100 and U25 = 115, so we can set up two equations using the formula above:

U20 = U1 + 19d = 100    

U25 = U1 + 24d = 115

We now have two equations with two variables (U1 and d) that we can solve for.

First, we'll isolate U1 in the first equation:

U1 = 100 - 19d

Then we'll substitute this expression for U1 into the second equation and solve for d:

100 - 19d + 24d = 115

5d = 15

d = 3

Substitute d = 3 in the equation, U1 = 100 - 19d

So, U1 = 100 - 19(3) = 43.

Therefore, the values of U and d for the arithmetic sequence are U= 43 and d = 3.

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The height of the replica souvenir is approximately 7.55 inches.

To find the height of the replica souvenir of the Statue of Liberty, we'll use the given scale factor of 1 inch = 20 feet. The original height of the statue is 151 feet. Divide the original height by the scale factor:

151 ft / 20 ft/in = 7.55 inches

The height of the replica souvenir is approximately 7.55 inches.

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Answer:

66.7%

Step-by-step explanation:

Let people asked bye x.

Then, people considering to change = 29% of x

People considering to become vegetarians = 23% of (29% of x)

                                                                          = 23/100 * 29x/100

                                                                          = 667x/10000

Percentage of people considering to become vegetarians = 667x/10

                                                                                                  = 66.7%

If two fair dice are rolled, the probability that the total showing is either even or less than five is 17/36. The correct answer is B.

There are 36 possible outcomes when rolling two dice, since each die has 6 possible outcomes.

To find the probability that the total showing is either even or less than five, we can first find the probability that the total showing is even, and then add to that the probability that the total showing is less than five, excluding the outcomes where the total showing is even.

To find the probability that the total showing is even, we can consider the following possibilities:

both dice show even numbers (probability 1/4)

both dice show odd numbers (probability 1/4)

So the probability of rolling an even total is 1/4 + 1/4 = 1/2.

To find the probability that the total showing is less than five, excluding the outcomes where the total showing is even, we can consider the following possibilities:

So the probability of rolling a total less than five, excluding the outcomes where the total showing is even, is 1/36 + 1/18 + 1/18 + 1/12 + 1/12 + 1/9 + 1/9 + 1/9 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 11/36.

Therefore, the probability that the total showing is either even or less than five is 1/2 + 11/36 = 17/36.

So the correct answer is (b) 17/36.

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The ratios that have a unit rate of 3 include the following:

A. 15/2 cups: 2 1/2 cups

C. 2 cups: 2/3 cups

F. 2 1/2 cups: 5/6 cups

In Mathematics, the unit rate is sometimes referred to as unit ratio and it can be defined as the quantity of material that is equivalent to a single unit of product or quantity.

15/2 cups : 5/2 cups

15/2 ÷ 5/2 : 5/2 ÷ 5/2

15/2 × 2/5  : 1

3 : 1  (True)

1 cup: 1/4 cups

1 × 4 : 1/4 × 1

4 : 1   (False)

2/3 cups: 1 cup

2/3 × 3/2  : 1 × 3/2

1 : 3/2   (False)

3 3/4 cups: 2 cups

(4 × 3 + 3)/4 : 2

15/4 : 2

15/8  : 2/2

15/8 : 1    (False).

2 cups: 2/3 cups

2 × 3/2 : 2/3 × 3/2

3  : 1   (True).

2 1/2 cups: 5/6 cups

(2 × 2 + 1)/2 : 5/6

5/2  : 5/6

5/2 × 6/5 : 5/6 × 6/5

3 : 1  (True).

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Complete Question:

Which ratios have a unit rate of 3? Choose all that apply.

15/2 cups: 2 1/2 cups

1 cup: 1/4 cups

2/3 cups: 1 cup

3 3/4 cups: 2 cups

2 cups: 2/3 cups

2 1/2 cups: 5/6 cups

The total number of cupcakes charity can make in 8 hours is 384

The total number of cupcakes she can make in 45 minutes is 36

Cupcakes she can make in 1 minute = 36/45

Cupcakes she can make in 1 minute = 0.8

Cupcakes she can make in 8 hours

We will convert hours into minutes

1 hour = 60 min

8 hour = 8 × 60 min

8 hour = 480 min

Cupcakes she can make in 8 hours that is 480 min = 480 × 0.8

Cupcakes she can make in 8 hours = 384

Total number of cupcakes she can make is 384

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Based on the information provided, the probability that Ben wins the board game is 30%.

To calculate the probability of Ben winning the board game, let's start by checking the information provided:

Probability for Justin to win: 20% or 0.2

Probability for Cam to win: 50% or 0.5

Now, the total probability is always equivalent to 100% or 0.1. Based on this, let's calculate now the probability that Ben wins the game.

1 - (0.2 + 0.5)  

1 - 0.7 = 0.3

Note: Here is the complete question:

Justin, Cam, and Ben are playing a board game where exactly one player will win. Ben estimates that Justin has a %20 percent chance of winning each game and that Cam has a %50 percent chance of winning each game. What is the probability that Ben will win the board game?

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Answer:

(3x+4) ÷ (x+6)

Step-by-step explanation:

3x²-14x-24 = (3x+4) (x-6)

x²-36 = (x+6) (x-6)

= (3x+4) (x-6) ÷ (x+6) (x-6)

Eliminate the (x-6)

= (3x+4) ÷ (x+6)

The system of equations can be used to determine the gallons of juice in container A and the gallon of juice in container B is  A + B = 28 and A + 6B = 88

Total amount of juice in both the vats is 28

Let the first container = A

second container = B

A + B = 28

Orange juice in A = 1/8 of A

Orange juice in B = 3/4 of B

Total orange juice in container A and B = 11

(1/8)A  + (3/4)B = 11

A + 6B = 88

Hence the two equations will be A + B = 28 and A + 6B = 88

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To find the volume and surface area of the composite figure, we need to first find the individual volumes and surface and the right cone, and then add them together.

The volume of a hemisphere with radius r is (2/3)πr^3, and the surface area is 2πr^2.

The volume of a right cone with radius r, height h, and slant s is (1/3)πr^2h, and the surface area is πr^2 + πrs.

In this case, the radius r and slant s are both 6 cm, and the height h of the cone can be found using the Pythagorean theorem: h = √(s^2 - r^2) = √(8^2 - 6^2) = √28 ≈ 5.29 cm.

So, the volume of the hemisphere is (2/3)π(6 cm)^3 = 72π/3 = 24π cubic cm, and the surface area is 2π(6 cm)^2 = 72π square cm.

The volume of the right cone is (1/3)π(6 cm)^2(5.29 cm) = 62.83π/3 ≈ 20.94π cubic cm, and the surface area is π(6 cm)^2 + π(6 cm)(8 cm) = 36π + 48π = 84π square cm.

Therefore, the total volume of the composite figure is 24π + 20.94π = 44.94π cubic cm, and the total surface area is 72π + 84π = 156π square cm.

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The component of the weight parallel to the inclined plane is approximately 34.72 pounds, and the component perpendicular to the inclined plane is approximately 196.96 pounds.

To analyze the situation, we need to break down the weight of the object into its components parallel and perpendicular to the inclined plane.

Given:

Weight of the object = 200 pounds

Angle of the inclined plane with the horizontal = 10 degrees

First, we find the component of the weight parallel to the inclined plane. This component can be determined using trigonometry:

Component parallel to the inclined plane = Weight * sin(angle)

Component parallel to the inclined plane = 200 pounds * sin(10 degrees)

Component parallel to the inclined plane ≈ 200 pounds * 0.1736

Component parallel to the inclined plane ≈ 34.72 pounds

Next, we find the component of the weight perpendicular to the inclined plane:

Component perpendicular to the inclined plane = Weight * cos(angle)

Component perpendicular to the inclined plane = 200 pounds * cos(10 degrees)

Component perpendicular to the inclined plane ≈ 200 pounds * 0.9848

Component perpendicular to the inclined plane ≈ 196.96 pounds

Therefore, the component of the weight parallel to the inclined plane  and the component perpendicular to the inclined plane is approximately 34.72 pounds and 196.96 pounds respectively.

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A telephone pole has a wire attached to its top that is anchored to the ground then conclude the height of the pole is approximately 51.53 feet.

Let h be the height of the pole. The equation h = (h - 69) + 6 represents the given information. Solving it gives h = 75.

Let's denote the height of the pole as "h". Then, according to the problem, the distance from the bottom of the pole to the anchor point is 69 feet less than the height of the pole, which means it is h - 69. Additionally, the wire is to be 6 feet longer than the height of the pole, so its length is h + 6.

Now we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (in this case, the wire) is equal to the sum of the squares of the lengths of the other two sides (in this case, the height of the pole and the distance from the bottom of the pole to the anchor point). So we have:

(h - 69)^2 + h^2 = (h + 6)^2

h^2 - 138h + 4761 + h^2 = h^2 + 12h + 36

h^2 - 75h - 1602 = 0

h = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = -75, and c = -1602.

h = (75 ± sqrt(75^2 - 4(1)(-1602))) / 2(1)

h ≈ 51.53 or h ≈ -31.53

Since the height of the pole cannot be negative, we can ignore the negative solution and conclude that the height of the pole is approximately 51.53 feet.

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The margin of error at the 99% confidence interval is 1.39%. Interpretation: we can be 99% confident that the true proportion of viewers under 30 in the population falls within the range of 30% ± 1.39%.

To calculate the margin of error at the 99% confidence interval, we can use the formula:

Margin of error = z* × √(p × (1 - p) / n)

where z* is the critical value (2.576 for a 99% confidence interval), p is the sample proportion (0.30), and n is the sample size (5000).

Margin of error = 2.576 × √(0.30 × (1 - 0.30) / 5000) ≈ 0.0139 or 1.39%

The interpretation of this result is that we can be 99% confident that the true proportion of viewers under 30 in the population falls within the range of 30% ± 1.39%. In other words, if we were to conduct the survey multiple times, we would expect the proportion of viewers under 30 to fall within this interval 99 out of 100 times. This information is useful for understanding the level of uncertainty in the survey results and can help guide decision-making based on the findings.

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The quadratic function in vertex form is y = (8/9)(x - 5)^2 + 7

The vertex form of a quadratic function is given by:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

In this case, we are given that the vertex is (5, 7), so we can write:

y = a(x - 5)^2 + 7

To find the value of a, we can use one of the points on the parabola.

Let's use the point (2, 15):

15 = a(2 - 5)^2 + 7

8 = 9a

a = 8/9

Substituting this value of a into the equation above, we get:

y = (8/9)(x - 5)^2 + 7

Therefore, the quadratic function in vertex form is y = (8/9)(x - 5)^2 + 7

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The mean is 35 minutes and the median is 34 minutes.

let's first calculate the mean, median, and mode for Sarah's exercise times: 34, 30, 45, 30, 40, 38, and 28 minutes.

Step 1: Calculate the mean
Add up all the values and divide by the total number of values:
(34 + 30 + 45 + 30 + 40 + 38 + 28) / 7 = 245 / 7 = 35 minutes (rounded)

Step 2: Calculate the median
Arrange the values in ascending order: 28, 30, 30, 34, 38, 40, 45
There are 7 values, so the median is the middle value: 34 minutes

Step 3: Calculate the mode
Determine the value(s) that occur most often: 30 minutes (occurs twice)

Now, should Sarah use the mean or the median to show she exercises for large amounts of time each day? The mean is 35 minutes and the median is 34 minutes. Both values are close and represent the central tendency of the data. However, since the mean is slightly higher than the median, Sarah should use the mean (35 minutes) to show she exercises for a larger amount of time each day.

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The artist used a total of $0.33 worth of paint for the project.

The artist purchased 2 liters of paint, which is equivalent to 2,000 milliliters of paint. This amount of paint was used to complete a project, and after the project was finished, there were 350 milliliters of paint left over.

To determine how much paint was used for the project, we subtract the amount of leftover paint from the total amount of paint purchased, which gives us 2,000 - 350 = 1,650 milliliters of paint used for the project.

The cost of the paint is 2 cents per milliliter, which is equivalent to $0.02/100 milliliters or $0.0002 per milliliter. To determine the cost of the paint used for the project, we multiply the amount of paint used by the cost per milliliter.

Therefore, the cost of 1,650 milliliters of paint used for the project can be calculated by multiplying 1,650 milliliters by $0.0002 per milliliter, which gives us $0.33.

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The area of the polygon composed of rectangles and triangle is 52 in²

Area is the amount of space occupied by a two dimensional shape or object.

For the first right triangle:

base = 2 in, height = 6 in

Area of first right triangle = 1/2 * base * height = 0.5 * 2 in * 6 in = 6 in²

For the second right triangle:

base = 2 in, height = 6 in

Area of second right triangle = 1/2 * base * height = 0.5 * 2 in * 6 in = 6 in²

For the triangle:

base = (2 + 4 + 2) = 8 in, height = 4 in

Area of triangle = 1/2 * base * height = 0.5 * 8 in * 4 in = 16 in²

For the rectangle:

length = 4 in, width = 6 in

Area of rectangle = length * width = 4 in * 6 in = 24 in²

Area of polygon = 6 + 6 + 16 + 24 = 52 in²

The area of the polygon is 52 in²

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Therefore, the linearization predicts that f(2.9, 4.1) is approximately 4.142.

To find the linearization of the function f(x, y) = √x^2 + y^2 at the point (3, 4), we need to find the partial derivatives of f with respect to x and y, evaluate them at (3, 4), and use them to write the equation of the tangent plane to the surface at that point.

First, we have:

∂f/∂x = x/√(x^2 + y^2)

∂f/∂y = y/√(x^2 + y^2)

Evaluating these at (3, 4), we get:

∂f/∂x(3, 4) = 3/5

∂f/∂y(3, 4) = 4/5

So the equation of the tangent plane to the surface at (3, 4) is:

z - f(3, 4) = (∂f/∂x(3, 4))(x - 3) + (∂f/∂y(3, 4))(y - 4)

Plugging in f(3, 4) = 5 and the partial derivatives, we get:

z - 5 = (3/5)(x - 3) + (4/5)(y - 4)

Simplifying, we get:

z = (3/5)x + (4/5)y - 1

This is the linearization of f(x, y) = √x^2 + y^2 at the point (3, 4).

To approximate f(2.9, 4.1), we plug in x = 2.9 and y = 4.1 into the linearization:

z = (3/5)(2.9) + (4/5)(4.1) - 1

z ≈ 4.142

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The police dog spends 1/6 of its workday at the police station.

To find the fraction of the police dog's workday spent at the police station, we need to add up the fractions of time spent in each location and subtract them from 1, since the dog spends the rest of the day at the police station.

Fraction of time spent in police car =  [tex]1/3[/tex]

Fraction of time spent in public = [tex]1/2[/tex]

To add these fractions, we need to find a common denominator:

[tex]1/3 = 2/6\\1/2 = 3/6[/tex]

So, the fraction of the dog's day spent at the police station is:

[tex]1 - (2/6 + 3/6) = 1 - 5/6[/tex]

                        = [tex]1/6[/tex]

Therefore, the police dog spends 1/6 of its workday at the police station.

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The height of the tree is approximately equal to the tangent of 768 radians multiplied by 50 meters, rounded to the nearest meter.

To find the height of the tree, we can use trigonometry. Let's assume that the height of the tree is represented by the variable "h."

In a right triangle formed by the forester, the base of the tree, and the top of the tree, the tangent of the angle is equal to the opposite side (height of the tree, h) divided by the adjacent side (distance from the base of the tree, 50 m).

Using the tangent function, we can write:

tan(angle) = [tex]\frac{h }{ 50}[/tex]

We can rearrange this equation to solve for h:

h = tan(angle) * 50

Now we can substitute the given angle into the equation and calculate the height:

h = tan(768) * 50

Calculating this in degrees might result in an error because the tangent function expects the angle to be in radians.

Therefore, we need to convert the angle to radians before evaluating the tangent.

There are 180 degrees in pi radians, so we can convert the angle as follows:

angle_radians = [tex]\frac{768 \* \pi }{ 180}[/tex]

Substituting this value into the equation:

h = tan(angle_radians) * 50

Now we can calculate the height using a calculator:

h ≈ [tex]tan(\frac{768 \pi }{ 180}) 50[/tex]

After evaluating this expression, we get the height of the tree. Rounding the answer to the nearest meter will give us the final result.

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Answer:

(-1/5)x + 8 = -x + 4

(4/5)x + 8 = 4

(4/5)x = -4

x = -5, so y = 9

C. There is one unique solution (-5, 9).

p(x) = x + 5, and k = 2. The expression x^2 + 7x + 12 / (x + 2) can be written in the form p(x) + k / (x + 2) as:

x + 5 + 2 / (x + 2)

To express the given expression x^2 + 7x + 12 / (x + 2) in the form p(x) + k / (x + 2), we will perform polynomial division.

1. Divide the numerator (x^2 + 7x + 12) by the denominator (x + 2):

(x^2 + 7x + 12) ÷ (x + 2)

2. Perform long division:

       x + 5
       ________________
x + 2 | x^2 + 7x + 12
       - (x^2 + 2x)
       ________________
             5x + 12
             - (5x + 10)
       ________________
                 2

3. Write the result:

p(x) + k / (x + 2) = x + 5 + 2 / (x + 2)

So, p(x) = x + 5, and k = 2. The expression x^2 + 7x + 12 / (x + 2) can be written in the form p(x) + k / (x + 2) as:

x + 5 + 2 / (x + 2)

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The work done to lift a 90 kg weight by 75 cm is approximately 661.5 joules.

Para calcular el trabajo realizado al elevar una pesa de 90 kg a una altura de 75 cm sobre el piso, necesitamos conocer la fuerza necesaria para levantar la pesa y la distancia que recorre la pesa para llegar a su altura máxima.

El trabajo se define como la energía necesaria para mover un objeto a través de una distancia determinada, y se calcula como el producto de la fuerza aplicada y la distancia recorrida.

En este caso, la fuerza necesaria para levantar la pesa es igual a su peso, que se calcula como su masa multiplicada por la aceleración debido a la gravedad (9.8 m/s²):

peso = masa x gravedad

    = 90 kg x 9.8 m/s²

    = 882 N

La distancia que recorre la pesa para elevarse 75 cm (o 0.75 m) es:

distancia = 0.75 m

Por lo tanto, el trabajo realizado al elevar la pesa es:

trabajo = fuerza x distancia

       = 882 N x 0.75 m

       = 661.5 J

Por lo tanto, se requiere un trabajo de 661.5 J para elevar una pesa de 90 kg a una altura de 75 cm sobre el piso.

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The divergence of the given vector field at all points where it's defined is div [tex]F = 4x - 2x × cos(x^2).[/tex]

To find the divergence of the given vector field at all points where it's defined, we will use the following terms:

divergence, vector field, and partial derivatives.

The given vector field is[tex]F = (2x^2 - sin(x^2)) i + 5j - sin(x^2) k.[/tex]

To find the divergence of F (div F), we need to take the partial derivatives of each component with respect to their

respective variables and then sum them up. So, div [tex]F = (∂(2x^2 - sin(x^2))/∂x) + (∂5/∂y) + (∂(-sin(x^2))/∂z)[/tex].

Find the partial derivative of the first component with respect to x:

[tex]∂(2x^2 - sin(x^2))/∂x = 4x - 2x × cos(x^2)[/tex] (applying chain rule).

Find the partial derivative of the second component with respect to y:

∂5/∂y = 0 (since 5 is a constant).

Find the partial derivative of the third component with respect to z:

[tex]∂(-sin(x^2))/∂z = 0[/tex] (since there is no z variable in the component).

Sum up the partial derivatives:

[tex]div F = (4x - 2x × cos(x^2)) + 0 + 0 = 4x - 2x × cos(x^2).[/tex]

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The line of regression equation for the mileage rating of a ( point) four wheel drive vehicle is [tex]\hat y = 16.111 + 0.365x,[/tex] and the best predicted new mileage rating of a ( point) four-wheel drive vehicle when x = 19 mi/gal, is equals to the 23.046 mi/gal. So, option(b) is right one.

A linear regression line has an equation of the form [tex]\hat y = a + bx,[/tex]

where x is the independent variable and y is the dependent variable. The slope of the line is b, and a is the estimated intercept (the value of y when x = 0). We have a table form data of old and new rating of four-wheel-drive vehicles. We have to determine the line of regression. Now, we have to calculate the value of 'a' and 'b'. Let the old and new mileage rating of four-wheel-drive vehicles be represented by vaiables 'x' and 'y'. Using the following formulas, [tex]b =\frac{ S_{xy}}{S_{xx}}[/tex] where, [tex]S_{xx} = \sum x² - \frac{ (\sum x)² }{n} [/tex]

where , [tex]\bar x = \frac{\sum x }{n}[/tex]

Here, n = 11, [tex]\sum x[/tex] = 235

[tex]\sum xy[/tex] = 263, [tex]\sum x²[/tex] = 5733, [tex]\sum xy[/tex] = 5879, so

[tex]S_{xx}[/tex] = 5733 - (235)²/11

= 5733 - 5020.454 = 712.546

[tex]S_{xy}[/tex] = 5879 - (235×263)/11

= 260.364

Now, b = 260.364/712.546 = 0.365

a = (263/11) - 0.365 ( 235/11)

= 23.909 - 7.798

= 16.111

So, regression line equation is

[tex]\hat y = 16.111 + 0.365x,[/tex]

The best predicted value of y, when x = 19 mi/gal, [tex]\hat y = 19× 0.365 + 16.111[/tex]

=23.046 mi/gal

Hence, the best predicted value is

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Complete question:

10. Determine the line of regression and use it to find the best predicted new mileage rating of a ( point) four-wheel-drive vehicle given that the old rating is 19 mi/gal

old 6 27 17 33 28 24 8 22 20 29 21

New 15 24 15 29 25 22 6 20 82 6 19

a) y cap = 0.863 + 0.808x; 16.2 mi/gal

b) y cap = 16.111 + 0.365x; 23.04 mi/gal

c) y cap =0.808+ 0.863x; 17.2 mi/gal

d) y cap = 0.808 0.863x; 18.1 mi/gal

The average walking rate for Path A is 2.5 mph, path B's is 2 mph and path C's is 3 mph.

Average walking rate = distance/time

for path A = 0.5/12 = 0.0416 mpm

To convert miles per meter into miles per hour it we will multiply the equation by 60 because 1 hour has 60 min

so, for path A = 0.0416 × 60 = 2.5 mph

Similarly , for path B = (1.5/45) × 60 = 2 mph

For path C = (0.75/15)×60 = 3 mph

Table to show the average walking rate

Path     Distance(mi)     Time(min)      Walking rate (mph)

  A              0.5                   12                    2.5

  B              1.5                    45                    2

  C              0.75                 15                     3

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None of the given options have a focus at (-6, 12) and directrix of x = -12,so none of the option is correct.

To find the equation with a focus at (-6, 12) and directrix of x = -12, we can use the general equation for a parabola with a vertical axis of symmetry:

(y - k)^2 = 4p(x - h)

where (h, k) is the focus and x = h - p is the directrix.

Given the focus (-6, 12) and directrix x = -12, we can determine the value of p:

p = h - (-12) = -6 - (-12) = 6

Now, we can plug in the values of h, k, and p into the equation:

(y - 12)^2 = 4(6)(x + 6)

Simplify the equation:

(y - 12)^2 = 24(x + 6)

Now, let's compare this equation to the given options:

1. (y - 12)^2 = 1/12 (x + 9)
2. (y - 12)^2 = -1/12 (x + 9)
3. (y - 12)^2 = 12 (x + 9)
4. (y - 12)^2 = -12 (x + 9)

None of the given options match the equation we found. Therefore, none of the given options have a focus at (-6, 12) and directrix of x = -12

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The statement "the series converges to 31.25" is true about the given series.

Given series is 25 + 5 + 1 + ....

We can clearly see that given series is infinite geometric series.

First term is a=25

common ratio is r = 5/25

= 1/5

We know that the formula of sum of an infinite geometric series is

S = a / (1 - r)

S = 25 / (1 - 1/5)

S = 25/(4/5)

S = (25*5)/4

S = 125/4

S = 31.5

Therefore, the sum of the infinite geometric series is 31.25.

Since, the sum of the series is a finite number, we can say that the series converges.

Therefore, the statement "the series converges to 31.25" is true about the given series.

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To evaluate the integral of (3x+6)/(2x²-3)dx, we can use partial fraction decomposition:

(3x+6)/(2x²-3) = A/(x-√(3)/2) + B/(x+√(3)/2)

Multiplying both sides by the denominator and simplifying, we get:

3x+6 = A(x+√(3)/2) + B(x-√(3)/2)

Setting x = √(3)/2, we get:

3√(3)/2 + 6 = B(√(3)/2-√(3)/2) = 0

So B = -2√(3). Setting x = -√(3)/2, we get:

-3√(3)/2 + 6 = A(-√(3)/2+√(3)/2) = 0

So A = 2√(3). Therefore, we have:

(3x+6)/(2x^2-3) = 2√(3)/(x-√(3)/2) - 2√(3)/(x+√(3)/2)

Integrating each term, we get:

∫(3x+6)/(2x²-3)dx = 2√(3)ln|x-√(3)/2| - 2√(3)ln|x+√(3)/2| + C

where C is the arbitrary constant.
To evaluate the integral of the function 3x + 6 with respect to x, we will use the integral symbol and find the antiderivative:

∫(3x + 6) dx

To find the antiderivative, we will apply the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the constant rule, which states that the integral of a constant is the constant times the variable:

(3 * (x^(1+1))/(1+1)) + (6 * x) + C

Simplifying the expression:

(3x²)/2 + 6x + C

Here, C is the arbitrary constant. So, the evaluated integral of 3x + 6 is:

(3x²)/2 + 6x + C

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