After a windstorm, a leaning pole makes a 75° angle with the road surface. the pole casts a 15-foot shadow when the sun is at a 45° angle of elevation. about how long is the pole?

The probability of getting to sleep in an interval is 0.0903.

The expected time the doctor spends awake in each interval is 1.8648 hours.

(a) To compute the long-run fraction of time the doctor spends sleeping, we can formulate a renewal reward process. In this process, each interval represents the time between consecutive emergencies.

Let T be the inter-arrival time between emergencies, which follows an exponential distribution with a rate of λ = 0.5 per hour. The average inter-arrival time is given by E(T) = 1/λ = 1/0.5 = 2 hours.

In each interval, the doctor can only get to sleep if it has been 36 minutes (6 hours) since the last emergency. Otherwise, she remains awake.

Let R be the reward obtained in each interval, which is the amount of time the doctor gets to sleep. If the doctor gets to sleep in an interval, the reward is (T - 0.6) since she has already waited for 0.6 hours (36 minutes). Otherwise, the reward is zero.

The long-run fraction of time spent sleeping, denoted by ρ, can be calculated as the expected reward per unit time:

ρ = E(R)/E(T)

To compute E(R), we need to consider the conditional probability that the doctor gets to sleep in an interval.

Given an interval length T, the probability that T > 0.1 (36 minutes) is given by P(T > 0.1) = 1 - P(T ≤ 0.1). This probability is equal to the cumulative distribution function (CDF) of the exponential distribution with rate λ evaluated at 0.1.

P(T > 0.1) = 1 - F(0.1) = 1 - (1 - exp(-λ * 0.1))

Substituting the value of λ = 0.5, we get:

P(T > 0.1) = 1 - (1 - exp(-0.5 * 0.1)) ≈ 0.0903

Therefore, the probability of getting to sleep in an interval is approximately 0.0903.

E(R) = (T - 0.6) * P(T > 0.1) + 0 * (1 - P(T > 0.1))

= (T - 0.6) * 0.0903

Substituting the average inter-arrival time E(T) = 2 hours:

E(R) = (2 - 0.6) * 0.0903 ≈ 0.1352 hours

Finally, we can compute ρ:

ρ = E(R)/E(T) = 0.1352/2 ≈ 0.0676

Therefore, the long-run fraction of time the doctor spends sleeping is approximately 0.0676.

(b) To compute E(ui), the expected time the doctor spends awake in each interval, we can use the fact that the total time spent in each interval is T, and the time spent sleeping is (T - R), where R is the reward obtained in each interval.

E(ui) = E(T - R)

= E(T) - E(R)

= 2 - 0.1352

≈ 1.8648 hours

Therefore, the expected time the doctor spends awake in each interval is approximately 1.8648 hours.

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Using the Pythagorean theorem, the smallest diameter of the pipe that will fit the fiber optic line is approximately 4.61 cm (Option C).

To determine the smallest diameter of the pipe that will fit the fiber optic line in rectangular casing BCDE, we need to find the diagonal AC of the rectangle. Since the rectangle is inscribed within circle A, the diameter of the circle will be equal to the diagonal of the rectangle.

Using the Pythagorean theorem, we can find the length of AC:

AC^2 = DE^2 + BE^2
AC^2 = (3 cm)^2 + (3.5 cm)^2
AC^2 = 9 + 12.25
AC^2 = 21.25
AC = √21.25 ≈ 4.61 cm

Therefore, the smallest diameter of the pipe that will fit the fiber optic line is approximately 4.61 cm (Option C).

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The value of angle Y in the pentagon is 139°.

The sum of the interior angles of a polygon can be found using the formula:

sum of  interior angles = (n - 2) * 180

where n is the number of sides of the polygon

A polygon with 5 sides is called pentagon. Thus, n = 5.

sum of  interior angles = (5 - 2)*180 = 540°

Thus,

∠U + ∠W + ∠X + ∠Y  + ∠Z = 540°

90 + 108 + 121 + ∠Y + 82 = 540

401 + ∠Y = 540

∠Y = 540 - 401

∠Y = 139°

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201 could refer to the mean of how many runners' times. The Option C is correct.

The sample of 32 runners, as given, does not refer to the mean of how many runners' times. The sample size refers to the number of individuals selected from the population while population size refers to the total number of individuals in the population.

Data:

The population of 201 people is given.

The sample of 32 runners is taken from the population.

So, the mean of the runners' times would be calculated using all 201 runners in the population, not just the 32 in the sample. Therefore, the Option C is correct.

Full question "If a sample of 32 runners were taken from a population of 201 runners, could refer to the mean of how many runners' times ? A. Both 32 and 201 B. Neither 32 nor 201 C. 201 D. 32"

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The table attached represents the pounds of each ingredient Amelia buys.

Let's denote the number of pounds of cashews that Amelia buys by "c", and the number of pounds of peanuts that she buys by "p".

According to the problem, Amelia buys 2 more pounds of peanuts than cashews. So, we have:

p = c + 2

Also, she buys 1 pound of dried fruit, which we can simply denote as "1".

The total bill for the snack mix is $41.11, so we can write:

0.5c + 0.75p + 1.5(1) = 41.11

where;

Simplifying the equation, we get:

0.5c + 0.75(c + 2) + 1.5 = 41.11

0.5c + 0.75c + 1.5 + 1.5 = 41.11

1.25c = 38.11

c = 30.488

Since we know that p = c + 2, we have:

p = 30.488 + 2 = 32.488

Now we can complete the table to show how many pounds of each ingredient Amelia buys:

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If you spin a spinner 75 times, the number of multiples of 2 could be either 7 or 38.

Assuming the spinner has an equal chance of landing on any number from 1 to 6, we can find the probability of landing on a multiple of 2 (2, 4, or 6) by dividing the number of multiples of 2 by the total number of possible outcomes:

Number of multiples of 2 = 3
Total number of possible outcomes = 6

So the probability of landing on a multiple of 2 is:

P(multiple of 2) = 3/6 = 1/2

This means that out of 75 spins, we can expect to land on a multiple of 2 about half the time. To find the exact number, we multiply the probability by the number of spins:

Number of multiples of 2 = P(multiple of 2) x Number of spins
Number of multiples of 2 = (1/2) x 75 = 37.5

Since we can't have a fraction of a spin, we need to round to the nearest whole number. In this case, we can round up or down depending on how we interpret the question. I

f we want to know how many times we can expect to land on a multiple of 2 on average, we should round down to 37.

If we want to know the closest integer to the expected value, we should round up to 38.

So depending on the context of the question, the answer could be either 37 or 38.

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Answer:

62.29

Step-by-step explanation:

100 - 3(4.25) - 13 - 4(2.99)

= 100 - 12.75 - 13 - 11.96

= 62.29

explanation in the picture

the answer is

62,29

The solution is, Options 1, 3, and, 5 are true.

The statements given are,

1.)   The product of reciprocals is 1.

Let the fraction be a/b , and reciprocal be b/a ,

The product of the two will be 1.

Hence, the statement is true.

2.)   To divide fractions, multiply the divisor by the reciprocal of the dividend.

Let the fraction be a/b , now,

a/b*1/a = a^2/b,

Hence, the statement is false.

3.)    The reciprocal of a whole number is 1 over the number.

Let the number be  3, now,

The reciprocal of number 3 is 1/3 .

Hence, the statement is true.

4.)   Reciprocals are used to multiply fractions.

The statement is false, Reciprocals are not used to multiply fractions.

5.)    To find the reciprocal of a fraction, switch the numerator and denominator.

Let the fraction be a/b , then the reciprocal will be b/a .

Hence, the statement is true.

Therefore, Options 1, 3, and, 5 are true.

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complete question:

Select all that apply. Determine which of the following statements are true of reciprocals. Select all that apply. The product of reciprocals is 1 To divide fractions, multiply the divisor by the reciprocal of the dividend The reciprocal of a whole number is 1 over the number Reciprocals are used to multiply fractions To find the reciprocal of a fraction, switch the numerator and denominator

The expression that represents the cost, in dollars, of admission to the museum including admittance to n special exhibits can be written as 12.50 + 2n

Here, 12.50 is the base admission cost without any special exhibit, and 2n represents the cost of n special exhibits, where each exhibit costs an extra $2.00.

By multiplying the number of special exhibits, n, by $2.00, we get the total cost of special exhibits, which we can then add to the base admission cost to get the total cost of admission to the museum including admittance to n special exhibits.

For example, if someone wants to visit the museum and see 3 special exhibits, the cost of admission would be:

12.50 + 2(3) = 12.50 + 6 = $18.50

Therefore, the expression 12.50 + 2n represents the total cost of admission to the museum including admittance to n special exhibits.

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Complete question is:

A museum charges $12.50 for admission. Each special exhibit costs an extra $2.00. Part A Write an expression that represents the cost, in dollars, of admission to the museum including admittance to n special exhibits

Answer: for d the answer is yes

Step-by-step explanation:

a line is in the form of y=mx+c or y=mx+b, as p(x)=3x+5 then the equation passes by being compatible with the general equation of a line

Step-by-step explanation:

1d.

yes.

f(x) = 2x + 3

g(x) = x + 2

p(x) = f(x) + g(x) = (2x + 3) + (x + 2) = 3x + 5

p(x) is still a linear function (highest exponent of x is 1). therefore it is a line.

Answer:

1,400

Step-by-step explanation:

its the same thing as 7 times 200

c. Yes, there is evidence of an association between grade level and preferred option to raise money.

a. The completed two-way frequency table summarizing the data on grade level and options to raise money is as follows:

         Car Wash | Bake Sale | Talent Show | Total

Seventh Graders[tex]| 66 | 15 | 16 | 98[/tex]

Eighth Graders [tex]| - | 50 | - | 50[/tex]

Total [tex]| 66 | 65 | 16 | 148[/tex]

Note: The "-" indicates that no data is available for those specific combinations.

b. To calculate the row relative frequencies, we divide each cell value by the corresponding row total and round to the nearest thousandth:

     Car Wash | Bake Sale | Talent Show

Seventh Graders [tex]| 0.673 | 0.153 | 0.163[/tex]

Eighth Graders [tex]| - | 1.000 | -[/tex]

Total [tex]| 0.446 | 0.439 | 0.115[/tex]

c. To determine if there is evidence of an association between grade level and preferred option to raise money, we can observe the row relative frequencies. If the relative frequencies differ substantially between the rows, it suggests an association. In this case, since the row relative frequencies for each option vary between the seventh and eighth graders, there is evidence of an association between grade level and the preferred option to raise money.

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Answer:

x = 15

I think thats what u wanted me to do- I hope it helps though. Middle school is hard.

Step-by-step explanation:

2x/3 + x/5 = 13

Here, we have to two different denominators in the LHS. So in order to solve the equation, we need to have like denominators and hence we find the Least Common Multiple (LCM).

The LCM of the denominators 3 and 5 is 15. Hence, we multiply each term to bring the denominator to 15.

5.(2x/3) + 3.(x/5) = 13

Now we combine the fractions with common denominator.

10x/15 + 3x/15 = 13

(10x + 3x)/15 =13

13x/15 = 13

Now, multiply the numbers and solve

13x = 13 . 15

x = 15 (cancelling 13 from both LHS and RHS)

The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a,b], and if M is any number between f(a) and f(b), then there exists at least one number c in the interval [a,b] such that f(c) = M.

In this case, we are given a continuous function g(x) with g(0) = 3, g(1) = 8, and g(2) = 4. Let s(x) be the inverse of g(x), which means that s(g(x)) = x for all x in the domain of g(x).

Suppose s(x) is invertible. Then for any y in the range of g(x), there exists a unique x such that g(x) = y, and therefore s(y) = x. In particular, let y = 5, which is between g(1) = 8 and g(2) = 4. By the Intermediate Value Theorem, there exists a number c in the interval [1,2] such that g(c) = 5.

However, this means that s(5) is not well-defined, since there are two values of x (namely c and s(5)) that satisfy g(x) = 5. Therefore, s(x) is not invertible.

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists a number c in the interval [a, b] such that f(c) = k.

Let g(x) be continuous with g(0) = 3, g(1) = 8, and g(2) = 4. Since g(x) is continuous, the Intermediate Value Theorem applies. However, to show that s(x) is not invertible, we need to show that g(x) is not one-to-one.

Notice that g(0) = 3 and g(2) = 4, with g(1) = 8 in between. This means that there must exist a point c1 in the interval (0, 1) such that g(c1) = 4, and another point c2 in the interval (1, 2) such that g(c2) = 3, due to the Intermediate Value Theorem.

Since g(c1) = g(c2) = 4 and c1 ≠ c2, g(x) is not one-to-one. Therefore, its inverse function s(x) does not exist, and s(x) is not invertible.

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Answer:

new perimeter of Brandy deck is 46 feet .

Step-by-step explanation:

The new perimeter of Brandy deck is 46 feet.

The entire length of all the sides of a rectangle is called the perimeter. As a result, we can calculate the perimeter of a rectangle by adding all four sides.

Using the given information,

Width = 7 Feet

Length = 12 feet

Perimeter = 2 (Width + Length)

[tex]= 2(7+12)[/tex]

[tex]=2(19)[/tex]

[tex]=38[/tex]

Perimeter when deck is 4 feet longer [tex]=38+ 4+4=46[/tex] Feet

Hence, the new perimeter of a deck is 46 feet.

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Answer:

Step-by-step explanation:

DE/EC.

Answer: The movie was 2 hours and 20 minutes long.

Step-by-step explanation:

basic adding + subtracting

The area of students photo with sides 10cm by 8cm is 80cm².

On order to calculate the area of a student's photo, one can use the formula for the area of a rectangle, which is:

Area = length x width. You can simply multiply the two numbers together to get the area of the photo in that unit squared (e.g. square centimeters).

If a student's photo has a length of 10 centimeters and a width of 8 centimeters, the area of the photo would be:

Area = 10 cm x 8 cm = 80 cm²

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What is the area of students photo with sides 10cm by 8cm

The mark-up value percentage is 25 %

Given data ,

The markup amount is the selling price minus the cost price, so:

Markup = $8630 - $6900 = $1730

The markup percentage is the markup amount divided by the cost price, expressed as a percentage:

Markup percentage = (Markup / Cost price) x 100%

Markup percentage = ($1730 / $6900) x 100%

Markup percentage = 0.25 x 100%

Markup percentage = 25%

Hence , the markup percentage is 25%

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The critical number A is 2

To find the critical number A for the function f(x) = 7(x-2)^(2/3):

We need to first find the derivative of the function and then set it equal to zero or identify where it is undefined.

Step 1: Find the derivative of f(x).
f'(x) = d/dx[7(x-2)^(2/3)]
Using the chain rule, we get:
f'(x) = (2/3) * 7(x-2)^(-1/3) * (1)

= (14/3)(x-2)^(-1/3)

Step 2: Set the derivative equal to zero or identify where it is undefined.
The derivative will never be zero since (14/3) is a constant and (x-2)^(-1/3) will never equal zero.

However, the derivative is undefined when the exponent -1/3 leads to a division by zero in the denominator.

This occurs when (x-2) = 0.

Solving for x, we get:
x-2 = 0
x = 2

Therefore, the critical number A is 2. The two important intervals for the function f(x) = 7(x-2)^(2/3) are (-∞, 2) and (2, ∞).

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From the margin gain formula, the number of hours he should have spent working on problems are equal to 2 hours and number of hours he should have spent reading are equal to other 2 hours. So, option (c) is correct one.

The meaning of ''margin'' is either the ''edge'' or the last unit and it is calculated by the incremental adjustment to the outcome, due to a unit change in the control variable. Marginal gain = Ratio of change in the outcome variable to the change in the control variable.

We have, total number of practice questions raises a students exam score

= 25

Along with practice questions the same amount as reading the textbook for 1 hour. In this problem, the outcome variable is the number of practice problems solved and the control variable is the number of hours Eric spent working on the practice problems. The above figure 1 table shows the total number of problems solved and time. First we determine the marginal gain of each hour Eric spent working on the practice questions. See the table present in above figures 2. Now, he has only 4 hours of study time for the best exam score as possible. We have to determine number of hours he should have spent reading and working on problems. It is assumed that students always cover the same number of pages during each hour they read the textbook so the advice provide by their teaching assistant that they can establish the relationship between time spent on reading the textbook and doing practice problems. The relation is below, 1 hour of reading the textbook = 15 practice problems solved. So, we can compare the the effectiveness of Eric's time spent on either working on practice problems or reading the textbook by using the table above figure 3. The decision rule for the optimal allocation of Eric's 4 hours of work is, If he can solve more than 15 practice problems in any of the 4 hours, then he should spend that particular hour working on the practice problems instead of reading the textbook. If not, then Eric should spend that hour to read the textbook instead.

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Complete question:

The table present in above figure 1 complete the question.

Eric is a hard-working college freshman. One Sunday, he decides to work nonstop until he has answered 100 practice problems for his math course. He starts work at 8:00 AM and uses a table to keep track of his progress throughout the day. He notices that as he gets tired, it takes him longer to solve each problem. later, the teaching assistant in charless economics course gives him some advice. the teaching assistant says, based on past experience, working on 25 problems raises a students exam score by about the same amount as reading the textbook for 1 hour. for simplicity, assume students always cover the same number of pages during each hour they spend reading. given this information, in order to use his 4 hours of study time to get the best exam score possible, how many hours should he have spent working on problems and how many should he have spent reading?

a) 1 hour working on problems, 3 hours reading

b) 2 hours working on problems, 2 hours reading

c) 3 hours working on problems, 1 hour reading

d) 4 hours working on problems, 0 hours reading

Answer:

c

Step-by-step explanation:

the new coordinates of the rhombus W'X'Y'Z' after a dilation with scale factor k=3 are: [tex]W'(3,0), X'(12,-3), Y'(15,-12), Z'(6,-9)[/tex]

To find the new coordinates of the image after dilation, we need to multiply the coordinates of each vertex by the scale factor k = 3.

Let's start with vertex W(1,0):

Multiply the x-coordinate by  [tex]3: 1 *\times 3 = 3[/tex]

Multiply the y-coordinate by [tex]3: 0 \times 3 = 0[/tex]

So the new coordinates of W' are [tex](3,0).[/tex]

Next, let's look at vertex X(4,-1):

Multiply the x-coordinate by [tex]3: 4 \times 3 = 12[/tex]

Multiply the y-coordinate by [tex]3: -1 \times 3 = -3[/tex]

So the new coordinates of X' are [tex](12,-3).[/tex]

Now for vertex Y(5,-4):

Multiply the x-coordinate by [tex]3: 5 \times 3 = 15[/tex]

Multiply the y-coordinate by [tex]3: -4 \times3 = -12[/tex]

So the new coordinates of Y' are  [tex](15,-12).[/tex]

Finally, let's consider vertex Z(2,-3):

Multiply the x-coordinate by  [tex]3: 2 \times 3 = 6[/tex]

Multiply the y-coordinate by  [tex]3: -3 \times3 = -9[/tex]

So the new coordinates of Z' are [tex](6,-9)[/tex]  .

Therefore, the new coordinates of the rhombus  [tex]W'X'Y'Z'[/tex] after a dilation with scale factor k=3 are:

[tex]W'(3,0)[/tex]

[tex]X'(12,-3)[/tex]

[tex]Y'(15,-12)[/tex]

[tex]Z'(6,-9)[/tex]

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Answer:

252

Step-by-step explanation:

So their are 38 more numbers to get to 41 and the numbers are adding by 6, so mulitply 6 by 38 and you get 228 and add 228 to the biggest number of 24 and your final answer becomes 252.

The percentage of females who do not satisfy the minimum score requirement of 925 on the SAT-I test is 35.9%.

Calculating the z-score for the minimum score requirement:
z = (X - Mean) / Standard Deviation
z = (925 - 998) / 202
z = -73 / 202 ≈ -0.361

Now, using the z-score to find the percentage of females below the minimum score:
Since the z-score is -0.361, we can use a z-table (or an online calculator) to find the area to the left of this z-score, which represents the percentage of females who scored below 925. The area to the left of -0.361 is approximately 0.359.

3. Convert the area to a percentage:
Percentage = Area * 100
Percentage = 0.359 * 100 ≈ 35.9%

So, approximately 35.9% of females do not satisfy the minimum score requirement of 925 on the SAT-I test.

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The triangle is a right triangle.

Any given triangle in which one of its internal angles measures 90^o is said to be a right angle. Thus there is a common relations among the three sides of the triangle which is shown by the Pythagorean theorem.

The Pythagorean theorem states that;

For a right triangle, this relation holds among its three sides;

/Hyp/^2 = /Adj/^2 + /Opp/^2

Considering the given diagram, we have;

/Hyp/^2 = /Adj/^2 + /Opp/^2

             = 4^2 + 3^2

             = 16 + 9

             = 25

so that;

Hyp = 25^1/2

       = 5

Therefore the given triangle is a right triangle.

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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 account balance after 2 years is approximately $107.15.

The compound interest formula is given by:

A = P * [tex](1 + r/n)^(^n^*^t^)[/tex]

Where:

P = Principal

r = Annual interest rate (as a decimal)

n = Number of times interest is compounded per year

t = Time in years

A = Final account balance

In this problem, we are given:

P = $100

r = 3.5% per year = 0.035 per year

n = 365 (since interest is compounded daily)

t = 2 years

Substituting these values in the formula, we get:

A = [tex]100 * (1 + 0.035/365)^(^3^6^5^*^2^)[/tex]

A ≈ $107.15

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The inequailty y < x + 3 would be true when x = -3 and all values of y are less than 1

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

The statement that x = -3

The above value implies that we substitute -3 for x in an inequality and solve for the variable y

Take for instance, we have

y < x + 4

Substitute the known values in the above equation, so, we have the following representation

y < -3 + 4

Evaluate

y < 1

This means that the inequailty y < x + 3 would be true when x = -3 and all values of y are less than 1

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The volume of the solid is π/20,

option (A). is correct.

Volume is described as  a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units.

we have that the  limits of integration for x are 0 and 1, because  the solid lies in the region between x = 0 and x = 1.

Hence, we can say that  the volume of the solid is given by:

V = ∫[0,1] (1/4)πx^4 dx

V = (1/4)π ∫[0,1] x^4 dx

V = (1/4)π (1/5) [x^5]0^1

V = (1/20)π

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