How to find the volume of a sphere - ACT Math

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Question

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate volume of the basketball? Remember that the volume of a sphere is calculated by V=(4πr3)/3

Answer

To find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. Then we would plug into the formula for volume V=(4π〖(4.7)〗3) / 3 (The information given of 22 ounces is useless)

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Question

If the diameter of a sphere is , find the approximate volume of the sphere?

Answer

The volume of a sphere = $\frac{4}{3}\pi r^{3}$

Radius is $\frac{1}{2}$ of the diameter so the radius = 5.

$V=\frac{4}{3}\pi5^{3}$

or $V=\frac{500}{3\pi}$

which is approximately $167\pi$

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Question

For a sphere the volume is given by V = (4/3)πr_3 and the surface area is given by A = 4_πr_2. If the sphere has a surface area of 256_π, what is the volume?

Answer

Given the surface area, we can solve for the radius and then solve for the volume.

4_πr_2 = 256_π_

4_r_2 = 256

_r_2 = 64

r = 8

Now solve the volume equation, substituting for r:

V = (4/3)π(8)3

V = (4/3)π*512

V = (2048/3)π

V = 683_π_

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Question

A cube has a side dimension of 4. A sphere has a radius of 3. What is the volume of the two combined, if the cube is balanced on top of the sphere?

Answer

$V_{cube}=(\text{side})^3=(4)^3=64$

$V{sphere}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3^3)=\frac{4}{3}\pi(27)=36\pi$

$V_{total}=64+36\pi$

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Question

What is the volume of a sphere with a diameter of 6 in?

Answer

The formula for the volume of a sphere is:

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

where = radius. The diameter is 6 in, so the radius will be 3 in.

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Question

The radius of a sphere is . What is the approximate volume of this sphere?

Answer

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

$V=\frac{4}{3}\pi(6)^3$

$V=\frac{4}{3}\pi(216)$

$V=288\pi$

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Question

What is the volume of a sphere with a diameter of inches? Leave your answer in terms of $\pi$ .

Answer

To find the volume of a sphere we use the sphere volume formula:

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

First we need to find the radius of the sphere. A sphere has a radius of half the diameter. So we see that .

Next we plug 6 in for our radius and get
$V = \frac{4}{3}\pi 6^3=288\pi in^3$

(don't forget your units).

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Question

What is the volume of a sphere with a diameter of $\textup{6 inches}$ (reduce all fractions)?

Answer

The formula for the volume of a sphere is:
$v = \frac{4}{3}\pi * r^3$ , thus we need to just determine the radius and plug it into the equation. Remember that and so

And plugging in we get $\frac{108}{3}\pi \textup{in}^3$

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Question

What is the volume of a sphere with a surface area of $576\pi\textup{ ft}^2$ ? (Simplify all fractions in your answer.)

Answer

First find the radius from the surface area set the given surface area equal to the surface area formula and solve for the radius.
$576\pi = 4\pi r^2$

$144\pi = \pi r^2 ewline 144 = r^2 ewline 12 = r$

Now plug the radius into the volume formula:
$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi 12^3 = 2304 \pi\textup{ ft}^3$

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Question

If Ariana’s orange has twice the radius of Autumn’s orange, the volume of Ariana’s orange is how many times larger than the volume of Autumn’s orange?

Answer

Define the radius of Autumn’s orange as r. The volume of her orange is $\pi r^3$ . Ariana’s orange has twice the radius of Autumn’s, so the radius of her orange is , and the volume is $\pi(2r)^3 = 8\pi r^3$ , which is 8 times larger than Autumn’s orange.

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Question

Find the volume of a sphere whose diameter is .

Answer

To solve, simply use the formula for the volume of a sphere. Thus,

$V=\frac{4}{3}\pi{r^3}=\frac{4}{3}\pi2^3=\frac{4}{3}8\pi=\frac{32}{3}\pi$

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Question

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate volume of the basketball? Remember that the volume of a sphere is calculated by V=(4πr3)/3

Answer

To find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. Then we would plug into the formula for volume V=(4π〖(4.7)〗3) / 3 (The information given of 22 ounces is useless)

Compare your answer with the correct one above

Question

If the diameter of a sphere is , find the approximate volume of the sphere?

Answer

The volume of a sphere = $\frac{4}{3}\pi r^{3}$

Radius is $\frac{1}{2}$ of the diameter so the radius = 5.

$V=\frac{4}{3}\pi5^{3}$

or $V=\frac{500}{3\pi}$

which is approximately $167\pi$

Compare your answer with the correct one above

Question

For a sphere the volume is given by V = (4/3)πr_3 and the surface area is given by A = 4_πr_2. If the sphere has a surface area of 256_π, what is the volume?

Answer

Given the surface area, we can solve for the radius and then solve for the volume.

4_πr_2 = 256_π_

4_r_2 = 256

_r_2 = 64

r = 8

Now solve the volume equation, substituting for r:

V = (4/3)π(8)3

V = (4/3)π*512

V = (2048/3)π

V = 683_π_

Compare your answer with the correct one above

Question

A cube has a side dimension of 4. A sphere has a radius of 3. What is the volume of the two combined, if the cube is balanced on top of the sphere?

Answer

$V_{cube}=(\text{side})^3=(4)^3=64$

$V{sphere}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(3^3)=\frac{4}{3}\pi(27)=36\pi$

$V_{total}=64+36\pi$

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Question

What is the volume of a sphere with a diameter of 6 in?

Answer

The formula for the volume of a sphere is:

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

where = radius. The diameter is 6 in, so the radius will be 3 in.

Compare your answer with the correct one above

Question

The radius of a sphere is . What is the approximate volume of this sphere?

Answer

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

$V=\frac{4}{3}\pi(6)^3$

$V=\frac{4}{3}\pi(216)$

$V=288\pi$

Compare your answer with the correct one above

Question

What is the volume of a sphere with a diameter of inches? Leave your answer in terms of $\pi$ .

Answer

To find the volume of a sphere we use the sphere volume formula:

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

First we need to find the radius of the sphere. A sphere has a radius of half the diameter. So we see that .

Next we plug 6 in for our radius and get
$V = \frac{4}{3}\pi 6^3=288\pi in^3$

(don't forget your units).

Compare your answer with the correct one above

Question

What is the volume of a sphere with a diameter of $\textup{6 inches}$ (reduce all fractions)?

Answer

The formula for the volume of a sphere is:
$v = \frac{4}{3}\pi * r^3$ , thus we need to just determine the radius and plug it into the equation. Remember that and so

And plugging in we get $\frac{108}{3}\pi \textup{in}^3$

Compare your answer with the correct one above

Question

What is the volume of a sphere with a surface area of $576\pi\textup{ ft}^2$ ? (Simplify all fractions in your answer.)

Answer

First find the radius from the surface area set the given surface area equal to the surface area formula and solve for the radius.
$576\pi = 4\pi r^2$

$144\pi = \pi r^2 ewline 144 = r^2 ewline 12 = r$

Now plug the radius into the volume formula:
$V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi 12^3 = 2304 \pi\textup{ ft}^3$

Compare your answer with the correct one above

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