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 feet, of a spherical tank with diameter 36 inches.

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 volume formula, and solve for :

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

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

$V = \frac{4}{3}\cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{1}{1}\cdot \frac{1}{1} \cdot \frac{3}{1} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{9}{2} \pi$

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Question

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

Answer

A sphere with radius has diameter and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

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Question

Which is the greater quantity?

(a) The volume of a cube with sidelength inches.

(b) The volume of a sphere with radius inches.

Answer

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius inches has diameter inches, and can therefore be inscribed inside the cube with sidelength inches. This give the cube larger volume, making (a) the greater quantity.

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Question

Which is the greater quantity?

(a) The volume of a sphere with diameter one foot

(b) $864 \textrm{ in}^{3}$

Answer

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute :

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

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

$V = \frac{4}{3} \pi \cdot 216$

$V = 288 \pi \approx 288 \cdot 3.14 = 904.32$ cubic inches,

which is greater than $864 \textrm{ in}^{3}$ .

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Question

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

Answer

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

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Question

Which is the greater quantity?

(a) The radius of a sphere with surface area $36 \pi$

(b) The radius of a sphere with volume $36 \pi$

Answer

The formula for the surface area of a sphere, given its radius , is

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

The sphere in (a) has surface area $36 \pi$ , so

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

$36 \pi\div 4 \pi = 4 \pi r^{2} \div 4 \pi$

$9 = r^{2}$

$r = \sqrt{9} = 3$

The formula for the volume of a sphere, given its radius , is

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

The sphere in (b) has volume $36 \pi$ , so

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

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

$27 \pi = \pi r^{3}$

$27 \pi\div \pi = \pi r^{3} \div \pi$

$27 = r^{3}$

$r = \sqrt[3]{27} = 3$

The radius of both spheres is 3.

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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.

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 $\pi$ ).

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?

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?

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?

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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Question

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

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.

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.

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.

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 $\frac{1}{4}$ .

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 feet, of a spherical tank with diameter 36 inches.

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 volume formula, and solve for :

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

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

$V = \frac{4}{3}\cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{1}{1}\cdot \frac{1}{1} \cdot \frac{3}{1} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{9}{2} \pi$

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Question

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

Answer

A sphere with radius has diameter and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

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Question

Which is the greater quantity?

(a) The volume of a cube with sidelength inches.

(b) The volume of a sphere with radius inches.

Answer

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius inches has diameter inches, and can therefore be inscribed inside the cube with sidelength inches. This give the cube larger volume, making (a) the greater quantity.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(a) The volume of a sphere with diameter one foot

(b) $864 \textrm{ in}^{3}$

Answer

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute :

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

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

$V = \frac{4}{3} \pi \cdot 216$

$V = 288 \pi \approx 288 \cdot 3.14 = 904.32$ cubic inches,

which is greater than $864 \textrm{ in}^{3}$ .

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