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slamgirl [31]
3 years ago
9

Find the volume of the solid whose base is the region in the first quadrant bounded by y=x3, y=1, and the y-axis and whose cross

-sections perpendicular to the x axis are semicircles.
Mathematics
1 answer:
Andrew [12]3 years ago
6 0

For a given x in the interval [0, 1], the corresponding cross-section is a semicircle with diameter d=1-x^3, so the area of the cross-section would be \frac{\pi d^2}8.

The volume of the solid is then

\displaystyle\frac\pi8\int_0^1(1-x^3)^2\,\mathrm dx=\boxed{\frac{9\pi}{112}}

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Step-by-step explanation:

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kipiarov [429]

Answer:

170 children

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85 adults

Step-by-step explanation:

Given

Let:

C = Children; S = Students; A = Adults

For the capacity, we have:

C + S + A = 329

For the tickets sold, we have:

5C + 7S + 12A = 2388

Half as many as adults are children implies that:

A = \frac{C}{2}

Required

Solve for A, C and S

The equations to solve are:

C + S + A = 329 -- (1)

5C + 7S + 12A = 2388 -- (2)

A = \frac{C}{2} -- (3)

Make C the subject in (3)

C = 2A

Substitute C = 2A in (1) and (2)

C + S + A = 329 -- (1)

2A + S + A = 329

3A + S = 329

Make S the subject

S = 329 - 3A

5C + 7S + 12A = 2388 -- (2)

5*2A + 7S + 12A = 2388

10A + 7S + 12A = 2388

7S + 22A = 2388

Substitute S = 329 - 3A

7(329 - 3A) + 22A = 2388

2303 - 21A + 22A = 2388

2303 +A = 2388

Solve for A

A = 2388 - 2303

A = 85

Recall that: C = 2A

C = 2 * 85

C = 170

Recall that: S = 329 - 3A

S = 329 - 3 * 85

S = 329 - 255

S = 74

Hence, the result is:

C = 170

S = 74

A = 85

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