How to find the volume of a sphere - ISEE Upper Level Quantitative Reasoning

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Question

In terms of $\pi$ , give the volume, in cubic inches, of a spherical water tank with a diameter of 20 feet.

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Answer

20 feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the volume formula, and solve for :

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

$V = $\frac{4}{3}$ \pi \cdot $120^{3}$$

$V = $\frac{4}{3}$ \pi \cdot 1,728,000$

$V = 2,304,000 \pi$ cubic inches

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Question

A sphere has diameter 3 meters. Give its volume in cubic centimeters (leave in terms of $\pi$ ).

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Answer

The diameter of 3 meters is equal to $3 \times 100 = 300$ centimeters; the radius is half this, or 150 centimeters. Substitute in the volume formula:

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

$V= $\frac{4}{3}$ \cdot \pi \cdot $150^{3}$$

$V= $\frac{4}{3}$ \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= $\frac{4}{3}$ \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= 4,500,000\pi$ cubic centimeters

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Question

A spherical buoy has a radius of 5 meters. What is the volume of the buoy?

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Answer

A spherical buoy has a radius of 5 meters. What is the volume of the buoy?

To find the volume of a sphere, use the following formula:

All we have to do is plug in 5 meters and simplify:

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Question

You have a ball with a radius of 12 cm, what is its volume?

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Answer

You have a ball with a radius of 12 cm, what is its volume?

The volume of a sphere can be found via the following formula:

We know our radius, so all we need to do is plug in and simplify:

So we have our answer:

$2304 \pi $cm^3$$

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Question

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the volume of the ball?

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Answer

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the volume of the ball?

Alright, let's begin with the volume of a sphere formula:

Now, plug in 12 and simplify:

$V_${sphere}=\frac{4}{3}$\pi(12in)^3$$=\frac{4}{3}$\pi*1728in^3$=2304\pi $in^3$$

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Question

Find the volume of a sphere with a diameter of 6cm.

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Answer

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 6cm. We also know that the diameter is two times the radius. Therefore, the radius is 3cm.

Knowing this, we can substitute into the formula. We get

$V = $\frac{4}{3}$ \cdot \pi \cdot $(3$\text{cm}$)^3$$

$V = $\frac{4}{3}$ \cdot \pi \cdot 3$\text{cm}$ \cdot 3$\text{cm}$ \cdot 3$\text{cm}$$

$V = $\frac{4}{3}$ \cdot \pi \cdot $27$\text{cm}$^3$$

Now, before we continue, we can simplify. The 3 and the 27 can both be divided by 3. We get

$V = $\frac{4}{1}$ \cdot \pi \cdot $9$\text{cm}$^3$$

$V = 4 \cdot \pi \cdot $9$\text{cm}$^3$$

$V = 36\pi $\text{ $cm}$^3$$

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Question

Find the volume of a sphere with a diameter of 12in.

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Answer

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in.

Knowing this, we can substitute into the formula. We get

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

$V = $\frac{4}{3}$ \cdot \pi \cdot 6$\text{in}$ \cdot 6$\text{in}$ \cdot 6$\text{in}$$

Now, we can simplify before we multiply to make things easier. The 3 and a 6 can both be divided by 3. So, we get

$V = $\frac{4}{1}$ \cdot \pi \cdot 2$\text{in}$ \cdot 6$\text{in}$ \cdot 6$\text{in}$$

$V = 4 \cdot \pi \cdot $72$\text{in}$^3$$

$V = 288\pi $\text{ $in}$^3$$

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Question

Find the volume of a sphere with a diameter of 6cm.

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Answer

To find the volume of a sphere, we will use the following formula

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 6cm. We also know the diameter is two times the radius. Therefore, the radius is 3cm.

So, we get

$V = $\frac{4}{3}$ \cdot \pi \cdot $(3$\text{cm}$)^3$$

$V = $\frac{4}{3}$ \cdot \pi \cdot $27$\text{cm}$^3$$

$V = $\frac{4}{3}$ \cdot $$\frac{27\text{cm}$^3$}{1} \cdot \pi$

$V = $\frac{4}{1}$ \cdot $$\frac{9\text{cm}$^3$}{1} \cdot \pi$

$V = $\frac{4 \cdot $9\text{cm}$^3$}{1 \cdot 1} \cdot \pi$

$V = $$\frac{36\text{cm}$^3$}{1} \cdot \pi$

$V = 36\pi $\text{ $cm}$^3$$

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Question

Find the volume of a sphere with a diameter of 18in.

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Answer

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 18in. We also know the diameter is two times the radius. Therefore, the radius is 9in.

Knowing this, we can substitute into the formula. We get

$V = $\frac{4}{3}$ \cdot \pi \cdot $(9$\text{in}$)^3$$

$V = $\frac{4}{3}$ \cdot \pi \cdot 9$\text{in}$ \cdot 9$\text{in}$ \cdot 9$\text{in}$$

Now, we can simplify before we multiply to make things easier. The 3 and a 9 can both be divided by 3. So, we get

$V = $\frac{4}{1}$ \cdot \pi \cdot 3$\text{in}$ \cdot 9$\text{in}$ \cdot 9$\text{in}$$

$V = 4 \cdot \pi \cdot $243$\text{in}$^3$$

$V = 972\pi $\text{ $in}$^3$$

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Question

Find the volume of a sphere with the radius of $\frac{1}{4}$ .

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Answer

Write the formula to find the volume of a sphere.

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

Substitute the radius into the formula.

$V=\frac{4}{3}$\pi $($\frac{1}{4}$)^3$$

Evaluate the equation.

$V=\frac{4}{3}$\pi ($\frac{1}{64}$) = $\frac{1}{3}$\pi ($\frac{1}{16}$) = $\frac{1}{48}$\pi$

The answer is: $$\frac{1}{48}$\pi$

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Question

In terms of $\pi$ , give the volume, in cubic inches, of a spherical water tank with a diameter of 20 feet.

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Answer

20 feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the volume formula, and solve for :

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

$V = $\frac{4}{3}$ \pi \cdot $120^{3}$$

$V = $\frac{4}{3}$ \pi \cdot 1,728,000$

$V = 2,304,000 \pi$ cubic inches

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Question

A sphere has diameter 3 meters. Give its volume in cubic centimeters (leave in terms of $\pi$ ).

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Answer

The diameter of 3 meters is equal to $3 \times 100 = 300$ centimeters; the radius is half this, or 150 centimeters. Substitute in the volume formula:

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

$V= $\frac{4}{3}$ \cdot \pi \cdot $150^{3}$$

$V= $\frac{4}{3}$ \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= $\frac{4}{3}$ \cdot 150 \cdot 150 \cdot 150\cdot \pi$

$V= 4,500,000\pi$ cubic centimeters

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Question

A spherical buoy has a radius of 5 meters. What is the volume of the buoy?

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Answer

A spherical buoy has a radius of 5 meters. What is the volume of the buoy?

To find the volume of a sphere, use the following formula:

All we have to do is plug in 5 meters and simplify:

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Question

You have a ball with a radius of 12 cm, what is its volume?

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Answer

You have a ball with a radius of 12 cm, what is its volume?

The volume of a sphere can be found via the following formula:

We know our radius, so all we need to do is plug in and simplify:

So we have our answer:

$2304 \pi $cm^3$$

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Question

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the volume of the ball?

Tap to reveal answer

Answer

You have a wooden ball which you are going to paint. If the radius is 12 inches, what is the volume of the ball?

Alright, let's begin with the volume of a sphere formula:

Now, plug in 12 and simplify:

$V_${sphere}=\frac{4}{3}$\pi(12in)^3$$=\frac{4}{3}$\pi*1728in^3$=2304\pi $in^3$$

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Question

Find the volume of a sphere with a diameter of 6cm.

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Answer

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 6cm. We also know that the diameter is two times the radius. Therefore, the radius is 3cm.

Knowing this, we can substitute into the formula. We get

$V = $\frac{4}{3}$ \cdot \pi \cdot $(3$\text{cm}$)^3$$

$V = $\frac{4}{3}$ \cdot \pi \cdot 3$\text{cm}$ \cdot 3$\text{cm}$ \cdot 3$\text{cm}$$

$V = $\frac{4}{3}$ \cdot \pi \cdot $27$\text{cm}$^3$$

Now, before we continue, we can simplify. The 3 and the 27 can both be divided by 3. We get

$V = $\frac{4}{1}$ \cdot \pi \cdot $9$\text{cm}$^3$$

$V = 4 \cdot \pi \cdot $9$\text{cm}$^3$$

$V = 36\pi $\text{ $cm}$^3$$

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Question

Find the volume of a sphere with a diameter of 12in.

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Answer

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in.

Knowing this, we can substitute into the formula. We get

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

$V = $\frac{4}{3}$ \cdot \pi \cdot 6$\text{in}$ \cdot 6$\text{in}$ \cdot 6$\text{in}$$

Now, we can simplify before we multiply to make things easier. The 3 and a 6 can both be divided by 3. So, we get

$V = $\frac{4}{1}$ \cdot \pi \cdot 2$\text{in}$ \cdot 6$\text{in}$ \cdot 6$\text{in}$$

$V = 4 \cdot \pi \cdot $72$\text{in}$^3$$

$V = 288\pi $\text{ $in}$^3$$

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Question

Find the volume of a sphere with a diameter of 6cm.

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Answer

To find the volume of a sphere, we will use the following formula

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 6cm. We also know the diameter is two times the radius. Therefore, the radius is 3cm.

So, we get

$V = $\frac{4}{3}$ \cdot \pi \cdot $(3$\text{cm}$)^3$$

$V = $\frac{4}{3}$ \cdot \pi \cdot $27$\text{cm}$^3$$

$V = $\frac{4}{3}$ \cdot $$\frac{27\text{cm}$^3$}{1} \cdot \pi$

$V = $\frac{4}{1}$ \cdot $$\frac{9\text{cm}$^3$}{1} \cdot \pi$

$V = $\frac{4 \cdot $9\text{cm}$^3$}{1 \cdot 1} \cdot \pi$

$V = $$\frac{36\text{cm}$^3$}{1} \cdot \pi$

$V = 36\pi $\text{ $cm}$^3$$

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Question

Find the volume of a sphere with a diameter of 18in.

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Answer

To find the volume of a sphere, we will use the following formula:

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

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 18in. We also know the diameter is two times the radius. Therefore, the radius is 9in.

Knowing this, we can substitute into the formula. We get

$V = $\frac{4}{3}$ \cdot \pi \cdot $(9$\text{in}$)^3$$

$V = $\frac{4}{3}$ \cdot \pi \cdot 9$\text{in}$ \cdot 9$\text{in}$ \cdot 9$\text{in}$$

Now, we can simplify before we multiply to make things easier. The 3 and a 9 can both be divided by 3. So, we get

$V = $\frac{4}{1}$ \cdot \pi \cdot 3$\text{in}$ \cdot 9$\text{in}$ \cdot 9$\text{in}$$

$V = 4 \cdot \pi \cdot $243$\text{in}$^3$$

$V = 972\pi $\text{ $in}$^3$$

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Question

Find the volume of a sphere with the radius of $\frac{1}{4}$ .

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Answer

Write the formula to find the volume of a sphere.

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

Substitute the radius into the formula.

$V=\frac{4}{3}$\pi $($\frac{1}{4}$)^3$$

Evaluate the equation.

$V=\frac{4}{3}$\pi ($\frac{1}{64}$) = $\frac{1}{3}$\pi ($\frac{1}{16}$) = $\frac{1}{48}$\pi$

The answer is: $$\frac{1}{48}$\pi$

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