Spheres - GRE Quantitative Reasoning

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Question

The surface area of a sphere is $36\pi$ . What is its diameter?

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Answer

The surface area of a sphere is defined by the equation:

$SA = 4\pi r^2$

For our data, this means:

$36\pi = 4\pi r^2$

Solving for , we get:

or

The diameter of the sphere is .

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Question

The volume of one sphere is $432\pi x^3$ . What is the diameter of a sphere of half that volume?

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Answer

Do not assume that the diameter will be half of the diameter of a sphere with volume of $432\pi x^3$ . Instead, begin with the sphere with a volume of $216\pi x^3$ . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

$216 \pi x^3=\frac{4}{3}\pi r^3$

Solving for , we get:

If you take the cube-root of both sides, you have:

$r = \sqrt[3]{162x^3}$

First, you can factor out an :

$r = x\sqrt[3]{162}$

Next, factor the :

$r = x\sqrt[3]{2*3^4}$

Which simplifies to:

$r = 3x\sqrt[3]{6}$

Thus, the diameter is double that or:

$6x\sqrt[3]{6}$

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Question

The surface area of a sphere is $36\pi$ . What is its diameter?

Tap to reveal answer

Answer

The surface area of a sphere is defined by the equation:

$SA = 4\pi r^2$

For our data, this means:

$36\pi = 4\pi r^2$

Solving for , we get:

or

The diameter of the sphere is .

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Question

The volume of one sphere is $432\pi x^3$ . What is the diameter of a sphere of half that volume?

Tap to reveal answer

Answer

Do not assume that the diameter will be half of the diameter of a sphere with volume of $432\pi x^3$ . Instead, begin with the sphere with a volume of $216\pi x^3$ . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

$216 \pi x^3=\frac{4}{3}\pi r^3$

Solving for , we get:

If you take the cube-root of both sides, you have:

$r = \sqrt[3]{162x^3}$

First, you can factor out an :

$r = x\sqrt[3]{162}$

Next, factor the :

$r = x\sqrt[3]{2*3^4}$

Which simplifies to:

$r = 3x\sqrt[3]{6}$

Thus, the diameter is double that or:

$6x\sqrt[3]{6}$

← Didn't Know|Knew It →

Question

The surface area of a sphere is $36\pi$ . What is its diameter?

Tap to reveal answer

Answer

The surface area of a sphere is defined by the equation:

$SA = 4\pi r^2$

For our data, this means:

$36\pi = 4\pi r^2$

Solving for , we get:

or

The diameter of the sphere is .

← Didn't Know|Knew It →

Question

The volume of one sphere is $432\pi x^3$ . What is the diameter of a sphere of half that volume?

Tap to reveal answer

Answer

Do not assume that the diameter will be half of the diameter of a sphere with volume of $432\pi x^3$ . Instead, begin with the sphere with a volume of $216\pi x^3$ . Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

$216 \pi x^3=\frac{4}{3}\pi r^3$

Solving for , we get:

If you take the cube-root of both sides, you have:

$r = \sqrt[3]{162x^3}$

First, you can factor out an :

$r = x\sqrt[3]{162}$

Next, factor the :

$r = x\sqrt[3]{2*3^4}$

Which simplifies to:

$r = 3x\sqrt[3]{6}$

Thus, the diameter is double that or:

$6x\sqrt[3]{6}$

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Question

What is the radius of a sphere with volume $36\pi$ cubed units?

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Answer

The volume of a sphere is represented by the equation $\frac{4}{3}\pi r^3$ . Set this equation equal to the volume given and solve for r:

$36\pi=\frac{4}{3}\pi r^3$

$36\pi\cdot \frac{3}{4}=\pi r^3$

$27\pi=\pi r^3$

Therefore, the radius of the sphere is 3.

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Question

If a sphere has a volume of cubic inches, what is the approximate radius of the sphere?

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Answer

The formula for the volume of a sphere is

$v=\frac{4}{3}(\pi r^3)$ where is the radius of the sphere.

Therefore,

$r=\sqrt[3]{\frac{3v}{(4\pi)}}$

$r=\sqrt[3]{\frac{3(268.08)}{(4\pi)}}=\sqrt[3]{63.999}=3.999$ , giving us $r\approx4$ .

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Question

A rectangular prism has the dimensions $9\textup{ x }6\textup{ x }12$ . What is the volume of the largest possible sphere that could fit within this solid?

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Answer

For a sphere to fit into the rectangular prism, its dimensions are constrained by the prism's smallest side, which forms its diameter. Therefore, the largest sphere will have a diameter of , and a radius of .

The volume of a sphere is given as:

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

And thus the volume of the largest possible sphere to fit into this prism is

$\frac{4}{3}\pi (3)^3=36\pi$

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Question

What is the radius of a sphere with volume $36\pi$ cubed units?

Tap to reveal answer

Answer

The volume of a sphere is represented by the equation $\frac{4}{3}\pi r^3$ . Set this equation equal to the volume given and solve for r:

$36\pi=\frac{4}{3}\pi r^3$

$36\pi\cdot \frac{3}{4}=\pi r^3$

$27\pi=\pi r^3$

Therefore, the radius of the sphere is 3.

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Question

If a sphere has a volume of cubic inches, what is the approximate radius of the sphere?

Tap to reveal answer

Answer

The formula for the volume of a sphere is

$v=\frac{4}{3}(\pi r^3)$ where is the radius of the sphere.

Therefore,

$r=\sqrt[3]{\frac{3v}{(4\pi)}}$

$r=\sqrt[3]{\frac{3(268.08)}{(4\pi)}}=\sqrt[3]{63.999}=3.999$ , giving us $r\approx4$ .

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Question

A rectangular prism has the dimensions $9\textup{ x }6\textup{ x }12$ . What is the volume of the largest possible sphere that could fit within this solid?

Tap to reveal answer

Answer

For a sphere to fit into the rectangular prism, its dimensions are constrained by the prism's smallest side, which forms its diameter. Therefore, the largest sphere will have a diameter of , and a radius of .

The volume of a sphere is given as:

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

And thus the volume of the largest possible sphere to fit into this prism is

$\frac{4}{3}\pi (3)^3=36\pi$

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Question

What is the radius of a sphere with volume $36\pi$ cubed units?

Tap to reveal answer

Answer

The volume of a sphere is represented by the equation $\frac{4}{3}\pi r^3$ . Set this equation equal to the volume given and solve for r:

$36\pi=\frac{4}{3}\pi r^3$

$36\pi\cdot \frac{3}{4}=\pi r^3$

$27\pi=\pi r^3$

Therefore, the radius of the sphere is 3.

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Question

If a sphere has a volume of cubic inches, what is the approximate radius of the sphere?

Tap to reveal answer

Answer

The formula for the volume of a sphere is

$v=\frac{4}{3}(\pi r^3)$ where is the radius of the sphere.

Therefore,

$r=\sqrt[3]{\frac{3v}{(4\pi)}}$

$r=\sqrt[3]{\frac{3(268.08)}{(4\pi)}}=\sqrt[3]{63.999}=3.999$ , giving us $r\approx4$ .

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Question

A rectangular prism has the dimensions $9\textup{ x }6\textup{ x }12$ . What is the volume of the largest possible sphere that could fit within this solid?

Tap to reveal answer

Answer

For a sphere to fit into the rectangular prism, its dimensions are constrained by the prism's smallest side, which forms its diameter. Therefore, the largest sphere will have a diameter of , and a radius of .

The volume of a sphere is given as:

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

And thus the volume of the largest possible sphere to fit into this prism is

$\frac{4}{3}\pi (3)^3=36\pi$

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Question

A sphere has a surface area of $16\pi$ square inches. If the radius is doubled, what is the surface area of the larger sphere?

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Answer

The surface area of the larger sphere is NOT merely doubled from the smaller sphere, so we cannot double $16\pi$ to find the answer.

We can use the surface area formula to find the radius of the original sphere.

$4\pi r^2=16\pi$

_r_2 = 4

r = 2

Therefore the larger sphere has a radius of 2 * 2 = 4.

The new surface area is then $4\pi \times 4^2=64\pi$ square inches.

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Question

Find the surface area of a sphere with a diameter of 14. Use π = 22/7.

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Answer

Surface Area = 4_πr_2 = 4 * 22/7 * 72 = 616

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Question

If a sphere has a volume of $36\pi$ cubic inches, what is its surface area?

Tap to reveal answer

Answer

The volume of a cube is equal to $v=\frac{4}{3}\pi r^3$ .

So we mutiply our volume by $\frac{3}{4}$ and divide by $\pi$ , giving us .

The surface area of a sphere is equal to $4\pi r^2$ , giving us $36\pi$ .

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Question

A sphere has a surface area of $16\pi$ square inches. If the radius is doubled, what is the surface area of the larger sphere?

Tap to reveal answer

Answer

The surface area of the larger sphere is NOT merely doubled from the smaller sphere, so we cannot double $16\pi$ to find the answer.

We can use the surface area formula to find the radius of the original sphere.

$4\pi r^2=16\pi$

_r_2 = 4

r = 2

Therefore the larger sphere has a radius of 2 * 2 = 4.

The new surface area is then $4\pi \times 4^2=64\pi$ square inches.

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Question

Find the surface area of a sphere with a diameter of 14. Use π = 22/7.

Tap to reveal answer

Answer

Surface Area = 4_πr_2 = 4 * 22/7 * 72 = 616

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