Spheres

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1

A sphere has surface area $64 \pi$ . Give its volume.

$\frac{256}{3}\pi$

$\frac{64}{3}\pi$

$64 \pi$

$128 \pi$

$256 \pi$

Explanation

The surface area of a sphere, given its radius , is equal to

$A= 4 \pi r^{2}$

Substitute $64\pi$ for and solve for :

$4 \pi r^{2} = 64 \pi$

Divide by $4 \pi$ :

$\frac{4 \pi r^{2}}{4 \pi} = \frac{64 \pi}{4 \pi}$

$r^{2} = 16$

Extract the square root:

Substitute for in the volume formula:

$V = \frac{4}{3} \pi r^{3}= \frac{4}{3} \pi \cdot 4^{3} =\frac{4}{3} \pi \cdot 64 = \frac{256}{3}\pi$ ,

the correct response.

2

The surface area of a sphere is equal to $100 \pi$ . Give the volume of the sphere.

$\frac{500}{3}\pi$

$500 \pi$

$\frac{250 }{3}\pi$

$250 \pi$

$\frac{1,000}{3}\pi$

Explanation

The surface area of a sphere can be calculated using the formula

$A = 4 \pi r^{2}$

Solving for :

Set $A = 100 \pi$ and divide both sides by $4 \pi$ :

$4 \pi r^{2} = 100 \pi$

$\frac{4 \pi r^{2}}{4 \pi} = \frac{100 \pi}{4 \pi}$

$r^{2} =25$

Take the square root of both sides:

$r = \sqrt{25} = 5$

Set in the volume formula:

$V= \frac{4}{3} \pi r^{3}$

$V= \frac{4}{3} \pi \left (5 \right )^{3} = \frac{4}{3} \pi \left (125 \right ) = \frac{500}{3}\pi$ ,

the correct response.

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