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Flashcards: Spheres

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

$250 \pi$ $\displaystyle 250 \pi$

$500 \pi$ $\displaystyle 500 \pi$

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

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

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

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Flashcards: Spheres

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

$250 \pi$ $\displaystyle 250 \pi$

$500 \pi$ $\displaystyle 500 \pi$

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

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

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

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Flashcards: Spheres

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

Your answer:

$250 \pi$ $\displaystyle 250 \pi$

Your answer:

$500 \pi$ $\displaystyle 500 \pi$

Your answer:

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

Your answer:

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

Correct answer:

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

Explanation:

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

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

Solving for $\displaystyle r$:

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

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

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

$r^{2} =25$ $\displaystyle r^{2} =25$

Take the square root of both sides:

$r = \sqrt{25} = 5$ $\displaystyle r = \sqrt{25} = 5$

Set r=5 $\displaystyle r=5$ in the volume formula:

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

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

the correct response.

Flashcards: Spheres

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

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

$64 \pi$ $\displaystyle 64 \pi$

$256 \pi$ $\displaystyle 256 \pi$

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

$128 \pi$ $\displaystyle 128 \pi$

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Flashcards: Spheres

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

Your answer:

$64 \pi$ $\displaystyle 64 \pi$

Your answer:

$256 \pi$ $\displaystyle 256 \pi$

Your answer:

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

Your answer:

$128 \pi$ $\displaystyle 128 \pi$

Correct answer:

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

Explanation:

The surface area $\displaystyle A$ of a sphere, given its radius $\displaystyle r$, is equal to

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

Substitute $64\pi$ $\displaystyle 64\pi$ for $\displaystyle A$ and solve for $\displaystyle r$:

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

Divide by $4 \pi$ $\displaystyle 4 \pi$:

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

$r^{2} = 16$ $\displaystyle r^{2} = 16$

Extract the square root:

r= 4 $\displaystyle r= 4$

Substitute for $\displaystyle r$ in the volume formula:

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

the correct response.

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