Spheres - ISEE Upper Level: Quantitative Reasoning

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Question

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

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Answer

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, set it equal to the given surface area and solve for r:

$900 \pi ft^2=4 \pi r^2$

First divide both sides by $4\pi$ .

Then square root both sides to get our radius:

Now, because the question is asking for our diameter and not our radius, we need to double our radius to get our answer:

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Question

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

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Answer

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4\pi r^2$

Now, plug in our surface area and solve with algebra:

$900 \pi m^2=4\pi r^2$

Get rid of the pi

Divide by 4

Square root both sides to get our answer:

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Question

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its radius?

Tap to reveal answer

Answer

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its radius?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, set it equal to the given surface area and solve for r:

$900 \pi ft^2=4 \pi r^2$

First divide both sides by $4\pi$ .

Then square root both sides to get our answer:

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Question

In terms of $\pi$ , give the surface area, in square feet, of a spherical tank with diameter 36 inches.

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Answer

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ , substitute in the surface area formula, and solve for :

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

$A =4\pi \cdot \left( \frac{3}{2} \right )^{2}$

$A =\frac{ 4}{1} \cdot \frac{3}{2}\cdot \frac{3}{2} \cdot \pi$

$A =\frac{ 1}{1} \cdot \frac{3}{1}\cdot \frac{3}{1} \cdot \pi$

$A =9 \pi$

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Question

Give the surface area of a sphere with diameter .

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Answer

A sphere with diameter has radius half that, or , so substitute into the formula for the surface area of a sphere:

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

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Question

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

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Answer

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

To find the surface area of a sphere, use the following:

$SA_{sphere}=4\pi r^2$

Plug in our radius and solve!

$SA_{sphere}=4\pi (5m)^2=100\pi m^2$

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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 surface area 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 surface area of the ball?

First, recall the formula for surface area of a sphere:

$SA_{sphere}=4 \pi r^2$

Now, just plug in our known radius and simplify:

$SA_{sphere}=4 \pi (12in)^2=576\pi in^2$

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Question

Find the surface area of a sphere with a diameter of 10in.

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Answer

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

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 10cm. We also know the diameter is two times the radius. Therefore, the radius is 5cm.

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

$SA = 4 \cdot \pi \cdot (5\text{cm})^2$

$SA = 4 \cdot \pi \cdot 25\text{cm}^2$

$SA = 100\text{cm}^2 \cdot \pi$

$SA = 100\pi \text{ cm}^2$

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Question

Find the surface area of a sphere with a radius of 10in.

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Answer

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

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 10in.

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

$SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400\pi \text{ in}^2$

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Question

Find the surface area of a sphere with a diameter of 18in.

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Answer

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

$SA = 4 \cdot \pi \cdot r^2$

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

$SA = 4 \cdot \pi \cdot (9\text{in})^2$

$SA = 4 \cdot \pi \cdot 81\text{in}^2$

$SA = 324 \pi \text{ in}^2$

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Question

Find the surface area of a sphere with a radius of 6in.

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Answer

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

$SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 6in. Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (6\text{in})^2$

$SA = 4 \cdot \pi \cdot 36\text{in}^2$

$SA = 144\pi \text{ in}^2$

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Question

Find the surface area of a sphere with a diameter of 12in.

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Answer

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

$SA = 4\pi r^2$

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. So, we get

$SA = 4 \cdot \pi \cdot (6\text{in})^2$

$SA = 4 \cdot \pi \cdot 36\text{in}^2$

$SA = 144\pi \text{ in}^2$

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Question

Find the surface area of a sphere with a radius of 12in.

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Answer

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

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 12in.

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

$SA = 4 \cdot \pi \cdot (12\text{in})^2$

$SA = 4 \cdot \pi \cdot 144\text{in}^2$

$SA = 576\pi \text{ in}^2$

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Question

Find the surface area of a sphere with a diameter of 20in.

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Answer

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

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 20in. We also know the diameter is two times the radius. Therefore, the radius is 10in.

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

$SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400 \pi \text{ in}^2$

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Question

A spherical buoy has a radius of meters. What is the surface area of the buoy?

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Answer

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

To find the surface area of a sphere, use the following:

$SA_{sphere}=4\pi r^2$

Plug in our radius and solve!

$SA_{sphere}=4\pi (5m)^2=100\pi m^2$

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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:

$V_{sphere}=\frac{4}{3}\pi r^3$

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

$V_{sphere}=\frac{4}{3}\pi (5m)^3=\frac{4}{3}\pi*125m^3=166\frac{2}{3}\pi m^3$

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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:

$V_{sphere}=\frac{4}{3}\pi r^3$

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

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

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:

$V_{sphere}=\frac{4}{3}\pi r^3$

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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