Find the distance from the plane 6x + 5y + z = 54 to the plane 6x + 5y + z = 48. The distance is d= (Type an exact answer, using radicals as needed.)

When we look at [tex]f(t)=1/7 t^3-4t-12[/tex], this is a cubic equation, and solving it analytically is not straightforward.

To find the value of r such that f(r) = 0, we need to solve the equation:

[tex]1/7 r^3 - 4r - 12 = 0[/tex]

This is a cubic equation, and solving it analytically is not straightforward.

Yet, it is possible to obtain the value of r that meets the equation using numerical schemes such as Newton-Raphson or bisection. Additionally, one can take advantage of calculation tools and graphical software to calculate an estimation of r.

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The required inequalities are h + l ≤ 20, 10h + 15l ≥ 240 and 20h + 25l ≥ 440

Given, for tutoring elementary students in math Susan earns $10 per hour. She earns $15 per hour for cleaning in the library.

Let h be the number of hours Susan works in one week tutoring elementary students.

Let l be the number of hours Susan works in one week cleaning the library.

Given that each week Susan cannot work more than 20 hours.

So, h + l ≤ 20 ....(1)

Susan's total earnings must be at least $240 per week.

10h + 15l ≥ 240    ...(2)

Multiplying equation (1) by 10

10h + 10l ≤ 200    ...(3)

Adding equations (2) and (3)

20h + 25l ≥ 440

Thus, the three required inequalities are h + l ≤ 20, 10h + 15l ≥ 240 and 20h + 25l ≥ 440

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The probability that the word so formed starts with M is 1/6, and the probability that it starts with M and ends with E is 1/30.

i) To find the probability that the word starts with M, we need to consider the total number of possible arrangements of the letters and the number of arrangements that start with M. The word "MOBILE" has 6 letters, so there are 6! = 720 possible arrangements of the letters. To find the number of arrangements that start with M, we can fix the M in the first position and arrange the remaining 5 letters in the remaining positions, which gives us 5! = 120 arrangements. Therefore, the probability that the word starts with M is:

P(starts with M) = number of arrangements that start with M / total number of arrangements

= 120 / 720

= 1/6

ii) To find the probability that the word starts with M and ends with E, we can fix the M in the first position and the E in the last position, and then arrange the remaining 4 letters in the remaining positions. This gives us 4! = 24 arrangements. Therefore, the probability that the word starts with M and ends with E is:

P(starts with M and ends with E) = number of arrangements that start with M and end with E / total number of arrangements

= 24 / 720

= 1/30

Thus, the probability that the word so formed starts with M is 1/6, and the probability that it starts with M and ends with E is 1/30.

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Since the area of a similar figure is proportional to the square of its linear dimensions, if one similar figure has an area that is nine times the area of another, the larger figure must have dimensions that are three times the dimensions of the smaller figure.

This is because the area is the square of the linear dimensions. So, if we increase the linear dimensions by a factor of 3, the area increases by a factor of 3^2 = 9.

Therefore, the answer is 3.

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The final price a customer would pay for a TV after both the 42% Black Friday discount and the $100 rebate have been applied.

Let x represent the original price of the TVs. The store is offering a Black Friday sale of 42% off, which we can represent as a function f(x) = 0.58x (since 100% - 42% = 58%). They are also giving a $100 rebate to all customers, represented by the function g(x) = x - 100.

Now, we want to find f(g(x)), which means applying the function f(x) to the result of the function g(x). So, f(g(x)) = f(x - 100).

To do this, plug in (x - 100) for x in the f(x) function: f(x - 100) = 0.58(x - 100).

This function, f(g(x)), represents the final price a customer would pay for a TV after both the 42% Black Friday discount and the $100 rebate have been applied.

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If A queen-sized mattress is 20 inches longer than it is wide. A king-sized mattress is 1280 square inches.

In mathematics, a variable is a symbol or letter that represents a value that can change or vary in a given context or problem. The area of the queen-sized mattress is x(x + 20) square inches. The equation to determine the area of the king-sized mattress is (x + 16)(x + 20) = x(x + 20) + 1280

Let x be the width of the queen-sized mattress in inches.

Then the length of the queen-sized mattress is x + 20 inches.

The length of the king-sized mattress is the same as the length of the queen-sized mattress, which is x + 20 inches.

We can use the formula for the area of a rectangle to find the area of each mattress:

Area of queen-sized mattress = length x width = (x + 20) x x = x^2 + 20x

Area of king-sized mattress = length x width = (x + 20) x (x + 16) = x^2 + 36x + 320

Area of king-sized mattress = Area of queen-sized mattress + 1,280

Substituting the expressions we found for the areas, we get:

x^2 + 36x + 320 = x^2 + 20x + 1280

Simplifying and solving for x, we get:

16x = 960

x = 60

So the width of the queen-sized mattress is 60 inches, and its length is 80 inches.

The width of the king-sized mattress is 76 inches, and its length is 80 inches.

The area of the queen-sized mattress is:

60^2 + 20(60) = 4,800 square inches

The area of the king-sized mattress is:

76^2 + 36(76) + 320 = 6,080 square inches

And we can verify that the area of the king-sized mattress is indeed 1,280 square inches more than that of the queen-sized mattress:

6,080 - 4,800 = 1,280

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The average price of milk in 2018 was $6.45 per gallon.

The average price of milk in 2021 was $189.15 per gallon.

When x = 0 (which represents the year 2018), the function becomes:

3.55 + 2.90(1 + 0)³

= 3.55 + 2.90(1)³

= 3.55 + 2.90

= 6.45

The average price of milk in 2018 was $6.45 per gallon.

When x = 3 (which represents the year 2021), the function becomes:

3.55 + 2.90(1 + 3)³

= 3.55 + 2.90(4)³

= 3.55 + 2.90(64)

= 3.55 + 185.6

= 189.15

The average price of milk in 2021 was $189.15 per gallon.

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Correspondence is a relation of connection as it establishes a link between two sets of elements, often by relating each element in one set to a specific element in the other set.

Correspondence refers to the exchange of communication or information between two or more parties. It is a relation of connection because it involves establishing a link or connection between the sender and the receiver of the message.

For example, when two people exchange letters or emails, they establish a correspondence that connects them and allows them to communicate. Similarly, in business, correspondence can refer to the exchange of official documents such as letters, memos, and reports, which establish a connection between different departments or organizations. Overall, correspondence is an important aspect of communication that helps to establish and maintain relationships between individuals and groups.

For example, in mathematics, a correspondence can be seen when matching the elements of one set to another, such as associating students with their grades. In this case, the connection is created by linking each student to their respective grade, illustrating the concept of correspondence.

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The explict formula, in slope intercept form is an = n/5

The given sequence is 1/5, 2/5, 3/5.

We can observe that this is an arithmetic sequence, where the first term is 1/5, the common difference is 1/5

To find the explicit formula for an arithmetic sequence, we can use the formula:

an = a1 + (n-1)d

Substituting the values we know for this sequence, we get:

an = 1/5 + (n - 1)*(1/5)

Evaluate

an = n/5

Thus, the nth term of this sequence can be found by substituting the value of n in the formula an = n/5

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Complete question

1/5 2/5 3/5

What is the explicit formula in slope intercept form

1- On the 58th birthday, the account would have $24,209.98, 2- If the interest is compounded monthly, then on the 58th birthday, the account would have $26,322.47.

1- The formula for calculating the compound interest is given by A = P(1 + r/n)(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years. Here, P = $10,000, r = 0.03, n = 1, t = 40 years (58 - 18).

substituting the values in the formula, we get A = $10,000(1 + 0.03/1)1*40) = $24,209.98.

2) In this case, n = 12 (monthly compounding), and t = 12*40 (total number of months in 40 years). So, the formula for calculating the compound interest becomes A = P(1 + r/n)(nt) = $10,000(1 + 0.03/12)(12*40) = $26,322.47.

Since the interest is compounded more frequently, the amount at the end of 40 years is higher than when the interest is compounded annually.

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The measure of angle PQR is 90 degrees.

Since PR is the diameter of the circle, it follows that angle POR is a right angle, i.e., it measures 90 degrees.

By the inscribed angle theorem, the measure of angle PQR is half the measure of angle POR. Thus,

angle PQR = 1/2 * angle POR

= 1/2 * 90

= 45 degrees.

However, this is not the final answer since angle PQR is not a stand-alone angle, but rather a part of a right-angled triangle PQR.

Since the three angles in a triangle add up to 180 degrees, and we already know that angle PQR is 45 degrees, it follows that:

angle PRQ + angle PQR + angle QPR = 180 degrees

Since angle PQR = 45 degrees, we have:

angle PRQ + 45 + angle QPR = 180 degrees

Rearranging, we get:

angle PRQ + angle QPR = 135 degrees

Since angles PRQ and QPR are complementary angles (together they form a right angle), their sum is 90 degrees. Therefore,

angle PRQ + angle QPR = 90 degrees

Substituting this into the previous equation, we get:

90 degrees = 135 degrees

This is a contradiction, and hence our assumption that angle PQR measures 45 degrees is false.

Therefore, we conclude that angle PQR must measure 90 degrees, since it is the only angle that can satisfy the given conditions.

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Answer:

When x is greater than -5, the expression inside the absolute value bars is positive, so we can simply remove the bars.

So the equation y = |x+5| can be rewritten as:

y = x+5 (when x > -5)

Over the last couple of years, Arnav's height has increased by 20% so his current height is 1.8 meters.

Arnav's height initially was 1.5 meters. Over the last couple of years, his height increased by 20%. To find the new height, we can use the formula: new height = initial height × (1 + percentage increase).

In this case, the initial height is 1.5 meters and the percentage increase is 20%, which can be expressed as a decimal (0.2). Using the formula, we can calculate Arnav's new height as follows:

New height = 1.5 meters × (1 + 0.2) = 1.5 meters × 1.2 = 1.8 meters.

After the 20% increase in height over the last couple of years, Arnav's current height is 1.8 meters.

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The distance between Town A and Town B is 550 km.

Let's denote the distance between Town A and Town B as D.

When the two cars first passed each other, let's assume that car A traveled a distance of x km and car B traveled a distance of D - x km.

Let's also denote the initial speeds of car A and car B as 6s and 5s, respectively, where s is some constant representing the speed of the slower car.

The time it took for the two cars to pass each other can be calculated using the formula:

time = distance / speed

For car A, the time it took to travel x km was:

x / (6s)

For car B, the time it took to travel D - x km was:

(D - x) / (5s)

Since the two cars traveled the same amount of time until they passed each other, we can set these two expressions equal to each other:

x / (6s) = (D - x) / (5s)

Solving for x, we get:

x = 6Ds / (11s)

After the speeds of both cars were reduced, car A's speed was (5/6) * 6s = 5s, and car B's speed was (3/4) * 5s = (15/4)s.

Let's denote the time it took for car A to travel the remaining distance from x to D as t.

Then, the time it took for car B to travel a distance of (D - x - 54) km is also t.

Using the new speeds, we can write the equation:

[tex](D - x - 54) = (15/4)s * t[/tex]

Solving for t, we get:

[tex]t = (4/15)(D - x - 54) / s[/tex]

The distance car A traveled after the two cars passed each other is:

D - x = D - 6Ds / (11s) = (5/11)D

The time it took for car A to travel this distance is:

[tex]t + x / (6s) = (4/15)(D - x - 54) / s + 6Ds / (66s)[/tex]

Setting these two expressions equal to each other and solving for D, we get:

D = 550 km

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The most general antiderivative of the function g(v) = 5 cos(v) - 8/√(1-v^2) is 5 sin(v) + 8 arcsin(v) + C, where C is the constant of the antiderivative.

To find the antiderivative of the given function g(v), we can use the basic antiderivative rules. The antiderivative of 5 cos(v) is 5 sin(v), as the derivative of sin(v) is cos(v) and we only need to reverse the process.

Similarly, the antiderivative of -8/√(1-v^2) can be found using the inverse trigonometric function arcsin(v), as its derivative is -1/√(1-v^2). However, we need to include a constant of integration, denoted by C, as the antiderivative is not unique.

So the most general antiderivative of g(v) is 5 sin(v) + 8 arcsin(v) + C, where C represents the constant of the antiderivative. To check the correctness of the answer, we can differentiate it and verify if it gives us the original function g(v) as the result.

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This will result in Circle 1 having the same center and radius as Circle 2, thus proving that the circles are similar.

To prove that Circle 1 and Circle 2 are similar, we can apply the following transformations to Circle 1:

1. Translation: Translate Circle 1 by moving its center from (-3, 5) to (7, 5). This is a horizontal translation of 10 units to the right.

2. Dilation: Dilate Circle 1 with a scale factor of 0.4, which will reduce its radius from 10 units to 4 units (the same as Circle 2).

These transformations will result in Circle 1 having the same center and radius as Circle 2, thus proving that the circles are similar.

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The length of MQ and QN is 10 and 15 respectively and verify that MQ: QN = 2:3

The coordinate of M = (-12,-5)

The coordinate of N = (8,10)

n = 2 , m = 3

By using the section formula coordinate of Q =( [tex]\frac{mx_{1} + nx_{2} }{m+n }[/tex] , [tex]\frac{my_{1} + ny_{2} }{m+n}[/tex])

Coordinate of Q = ([tex]\frac{(-12)3 + 8(2)}{3+2}[/tex] , [tex]\frac{10(2) + 3(-5)}{2+3}[/tex])

Coordinate of Q = ( -4, 1)

Now using the distance formula

MQ = [tex]\sqrt{ (x_{2}- x_{1} )^{2} +(y_{2} -y_{1} )^{2}[/tex]

MQ = [tex]\sqrt{(-4+12)^{2}+(1+5)^{2} }[/tex]

MQ = √100

MQ = 10

Similarly,

QN = [tex]\sqrt{(8+4)^{2}+(10-1)^{2} }[/tex]

QN =  [tex]\sqrt{225}[/tex]

QN = 15

MQ:QN = 10:15

MA :QN = 2:3

Hence it is verified that MQ: QN = 2:3

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The total or actual time he needs to practice is 140 min whereas he practiced for 98 min during the school week.

We need to find the total time he must practice for a week. To find the total time we assume that the total time is x min.

Given Data:

Dmitri practices time during the school week = 98 min

Dmitri practices amount of time = 70% of his total time

Total time = x

Then the equation is given as

70% × (x) = 98

0.70 × (x) = 98

x = 98 / 0.70

x = 140

Therefore, The total time of the practices is 140 min

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Answer:

1,215,000 cubic centimeters

Step-by-step explanation:

1. Find the volume of the cylinder

v = π r (squared) x h

v = 3.14 x 50 (squared) x 100

v = 3.14 x 2,500 x 100

v = 3.14 x 250,00

v = 785,000 cubic centimeters

2. Find the volume of the rectangular prism

v = l x w x h

v = 100 x 200 x 100

v = 2,000,000 cubic centimeters

3. Subtract

2,000,000 - 785,000 = 1,215,000 cubic centimeters

Answer:

Step-by-step explanation:

cat

The coordinates of the triangle after the dilation are given as follows:

a(0, 12), b(-6, 15) and c(9, 21).

The perimeter of the triangle increases, as the side lengths are multiplied by 3, hence the perimeter is also multiplied by 3.

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The scale factor for this problem is given as follows:

k = 3.

The scale factor is greater than 1, meaning that the figure is an enlargement, and thus the perimeter increases.

The original vertices of the triangle are given as follows:

a(0, 4), b(−2, 5), and c(3, 7)

Hence the vertices of the dilated triangle are given as follows:

a(0, 12), b(-6, 15) and c(9, 21).

(each coordinate of each vertex is multiplied by the scale factor of 3).

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The derivative of the given function with respect to x is:
f'(x) = e^(x^2 + 1) * (1 + 2x^2)

To find the derivative of the given function. Let's first rewrite the function for clarity: f(x) = x * e^(x^2 + 1).

To find the derivative f'(x) with respect to x, we'll apply the product rule since we have a product of two functions: x and e^(x^2 + 1). The product rule states that if you have a function f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).

In this case, g(x) = x and h(x) = e^(x^2 + 1). First, let's find the derivatives g'(x) and h'(x):

g'(x) = d/dx (x) = 1
h'(x) = d/dx (e^(x^2 + 1)) = e^(x^2 + 1) * d/dx (x^2 + 1) = e^(x^2 + 1) * (2x)

Now, we can apply the product rule:

f'(x) = g'(x) * h(x) + g(x) * h'(x) = 1 * e^(x^2 + 1) + x * (e^(x^2 + 1) * 2x)

Simplifying the expression, we get:

f'(x) = e^(x^2 + 1) + 2x^2 * e^(x^2 + 1)

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The statement that is true is that the amount of rain in the 24- hour period is 8 inches

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

In 90 minutes 1/2 inch of rain falls

This means that

Rate = (1/2 inch)/90 minutes

So, we have

Rate = (1/2 inch)/(1.5 hour)

The amount of rain in the 24- hour period is

Amount = Rate * Time

So, we have

Amount = (1/2 inch)/(1.5 hour) * 24 hours

Evaluate

Amount = 8 inches

Hence, the amount of rain in the 24- hour period is 8 inches

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The rate of change of the height of the top of the ladder is -144/h ft/sec when the base of the ladder is 6 ft from the house.

The area of the triangle formed by the ladder, wall, and ground is decreasing at a rate of 163.2 ft^2/sec when the base of the ladder is 6 ft from the house.

The angle between the ladder and the ground is decreasing at a rate of 1/8 rad/sec when the base of the ladder is 6 ft from the house.

The rate of change of the height of the top of the ladder, we need to use the Pythagorean Theorem:

where h is the height of the top of the ladder, d is the distance of the base of the ladder from the house, and L is the length of the ladder.

Taking the derivative with respect to time, t, and using the chain rule, we get:

We are given that d = 6 ft, dd/dt = 24 ft/sec, and L = 10 ft. We need to find dh/dt when d = 6 ft.

Plugging in the values, we get:

2h (dh/dt) + 2(6)(24) = 2(10) (0) (since the ladder is not changing length)

Simplifying, we get:

Dividing by 2h, we get:

The area of the triangle formed by the ladder, wall, and ground is given by:

where b is the distance of the base of the ladder from the wall, and h is the height of the triangle.

Taking the derivative with respect to time, t, and using the product rule, we get:

We are given that db/dt = -24 ft/sec, h = L, and dh/dt = -144/h. We need to find dA/dt when d = 6 ft.

Plugging in the values, we get:

Simplifying, we get:

The rate of change of the angle between the ladder and the ground, we use the trigonometric identity:

Dividing by sec^2(theta), we get:

d(theta)/dt = (-24/h^3) - (2h^2/5)

We can plug in the value of h = (L^2 - d^2)^(1/2) = (100 - 36)^(1/2) = 8 ft when d = 6 ft to get:

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The height of the cone is approximately 12.15 millimeters (rounded to the nearest hundredth).

To find the height of the cone, we need to use the formula for the volume of a cone:

V = (1/3)πr²h

where V is the volume, r is the radius, h is the height, and π is approximately equal to 3.14.

We are given the volume of the cone as 2,279.64 cubic millimeters. We can plug this value into the formula and solve for h:

2,279.64 = (1/3)πr²h

Multiplying both sides by 3 and dividing by πr², we get:

h = (3 × 2,279.64) / (π × r²)

Now, we need to find the radius of the cone. Unfortunately, we are not given this information directly. However, we can use the fact that the volume of a cone is also given by:

V = (1/3)πr²h

If we rearrange this formula to solve for r², we get:

r² = 3V / (πh)

Now, we can substitute the given values for V and h and simplify:

r² = 3(2,279.64) / (π × h) ≈ 2,304.32 / h

Taking the square root of both sides, we get:

r ≈ √(2,304.32 / h)

Now, we can substitute this expression for r into our earlier formula for h:

h = (3 × 2,279.64) / (π × r²) ≈ (6,838.92 / π) / (2,304.32 / h)

Simplifying, we get:

h ≈ 2,279.64 × h / (2,304.32 / h)

h² ≈ 2,279.64 × h / (2,304.32 / h)

h³ ≈ 2,279.64

Taking the cube root of both sides, we get:

h ≈ 12.15

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The area of the table is approximately 651 square inches.

What is Area ?

Area is a measure of the size of a two-dimensional shape or surface, such as a rectangle, circle, or triangle. It is expressed in square units, such as square inches, square feet, or square meters.

Let's use the Pythagorean theorem to find the length of the table top:

Substituting the given values, we get:

40*40 = [tex]length^{2}[/tex] + 21*21

Simplifying and solving for length, we get:

[tex]length^{2}[/tex]= 1600 - 441

[tex]length^{2}[/tex] = 961

length = 31 inches (rounded to the nearest inch)

Now that we know the length and width of the table, we can find the area by multiplying them together:

area = length x width

area = 31 x 21

area = 651 square inches (rounded to the nearest inch)

Therefore, the area of the table is approximately 651 square inches.

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Answer:

T=64

Step-by-step explanation:

Multiply both sides by 4

t/4=16

t/4×4=16×4 Cancel out the 4

t=64

a) The distance the ball rebounds on the fifth bounce is approximately 7.59 ft.

b) The total distance the ball has traveled after the fifth bounce is approximately 52.61 ft.

Let's denote the height of the ball after its nth bounce by h_n. Then we can express the relationship between the height of the ball after each bounce in terms of a recursive formula:

h_0 = 16 (initial height)

h_1 = (3/4) * h_0 (rebound distance after the first fall)

h_2 = (3/4) * h_1 (rebound distance after the second fall)

h_3 = (3/4) * h_2 (rebound distance after the third fall)

h_4 = (3/4) * h_3 (rebound distance after the fourth fall)

h_5 = (3/4) * h_4 (rebound distance after the fifth fall)

a) To find the distance the ball rebounds on the fifth bounce, we need to calculate h_5:

h_5 = (3/4) * h_4

= (3/4) * ((3/4) * ((3/4) * ((3/4) * 16)))

= (3/4)^5 * 16

= 7.59375 ft

Therefore, the ball rebounds approximately 7.59 ft on the fifth bounce.

b) To find the total distance the ball has traveled after the fifth bounce, we need to add up all of the distances traveled during the falls and rebounds:

total distance = distance of first fall + rebound distance after first fall + rebound distance after second fall + rebound distance after third fall + rebound distance after fourth fall + rebound distance after fifth fall

total distance = 16 + (3/4) * 16 + (3/4)^2 * 16 + (3/4)^3 * 16 + (3/4)^4 * 16 + (3/4)^5 * 16

total distance = 16 + 12 + 9 + 6.75 + 5.0625 + 3.7969

total distance = 52.6094 ft

Therefore, the ball travels approximately 52.61 ft after the fifth bounce.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

Be sure to show and explain all work using mathematical formulas and terminology. A bouncy ball is dropped from a height of 16ft and always rebounds ¼ of the distance of the previous fall.

a) What distance does it rebound the 5th time?

b) What is the total distance the ball has travelled after this time?