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

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Question

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

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Answer

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = $\frac{18}{12}$ = $\frac{3}{2}$$ feet

Radius: 4 inches = $4 \div 12 = $\frac{4}{12}$ = $\frac{1}{3}$$

The volume of a cone is given by the formula

$V = $\frac{1}{3}$ \pi $r^{2}$ h$

Substitute $r = $\frac{1}{3}$, h = $\frac{3}{2}$$ :

$V = $\frac{1}{3}$\cdot \left ( $\frac{1}{3}$ \right ) ^{2}\cdot $\frac{3}{2}$ \cdot \pi$

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

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

$V = $\frac{1}{18}$ \pi$

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Question

A cone has height 240 centimeters; its base has radius 80 centimeters. Give its volume in cubic meters.

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Answer

Convert both dimensions from centimeters to meters by dividing by 100:

Height: 240 centimeters = $240 \div 100 = 2.4$ meters.

Radius: 80 centimeters = $80 \div 100 = 0.8$ meters.

Substitute in the volume formula:

$V = $\frac{1}{3}$ \pi $r^{2}$h$

$V = $\frac{1}{3}$ \cdot \pi \cdot 0.8 ^{2} \cdot 2.4$

$V = $\frac{1}{3}$\cdot 0.8 \cdot 0.8 \cdot 2.4 \cdot \pi$

$V =0.512 \pi$

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Question

Give the volume of a cone whose height is 10 inches and whose base is a circle with circumference $6 \pi$ inches.

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Answer

A circle with circumference $6 \pi$ inches has as its radius

$r = $\frac{C }{2\pi }$=\frac{6\pi }{2\pi }$ = 3$ inches.

The area of the base is therefore

$B = \pi $r^{2}$ = \pi \cdot $3^{2}$ = 9 \pi$ square inches.

To find the volume of the cone, substitute $B = 9 \pi , h = 10$ in the formula for the volume of a cone:

$V = $\frac{1}{3}$ Bh = $\frac{1}{3}$ \cdot 9 \pi \cdot 10 = 30 \pi$ cubic inches

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Question

The height of a cone and the radius of its base are equal. The circumference of the base is $10 \pi$ inches. Give its volume.

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Answer

A circle with circumference $10 \pi$ inches has as its radius

$r = $\frac{C }{2\pi }$=\frac{10\pi }{2\pi }$ = 5$ inches.

The height is also inches, so substitute in the volume formula for a cone:

$V = $\frac{1}{3}$ \pi r ^{2}h = $\frac{1}{3}$ \pi \cdot 5 ^{2} \cdot 5 = $\frac{125}{3}$ \pi$ cubic inches

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Question

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

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Answer

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

Begin by using the formula for volume of a cone:

$V=\pi $r^2$$\frac{h}{3}$$

Now, we simply need to plug in our knowns.

We know the height is 30 ft.

We know that the diameter is 10ft, however, we need the radius.

Divide 10 by 2 to get a radius of 5 ft.

Now, let's go....

$V=\pi $(5ft)^2$$$\frac{30ft}{3}$=\pi25ft^2$*10ft=250 \pi $ft^3$$

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Question

Find the volume of a cone with the following measurements:

diameter = 12in

height = 6in

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Answer

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

$V = \pi $r^2$ $\frac{h}{3}$$

where r is the radius and h is the height of the cone.

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

We know the height of the cone is 6in.

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

$V = \pi \cdot $(6$\text{in}$)^2$ \cdot $\frac{6\text{in}$}{3}$

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

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

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

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Question

Find the volume of a cone with the following measurements:

height: 12in

diameter: 6in

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Answer

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

$V = \pi $r^2$ $\frac{h}{3}$$

where r is the radius and h is the height of the cone.

Now, we know the height of the cone is 12in. We also know the diameter of the cone is 6in. We know the diameter is two times the radius. Therefore, the radius is 3in.

So, we get

$V = \pi \cdot $(3$\text{in}$)^2$ \cdot $\frac{12\text{in}$}{3}$

$V = \pi \cdot $9$\text{in}$^2$ \cdot 4$\text{in}$$

$V = \pi \cdot $36$\text{in}$^3$$

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

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Question

Find the volume of a cone with the following measurements:

Diameter: 14in
Height: 9in

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Answer

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

$V = \pi $r^2$ $\frac{h}{3}$$

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 14in. We also know the diameter is two times the radius. Therefore, the radius is 7in.

We know the height of the cone is 9in.

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

$V = \pi \cdot $(7$\text{in}$)^2$ \cdot $\frac{9\text{in}$}{3}$

$V = \pi \cdot $49$\text{in}$^2$ \cdot 3$\text{in}$$

$V = \pi \cdot $147$\text{in}$^3$$

$V = 147\pi $\text{ $in}$^3$$

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Question

Find the volume of a cone with a base diameter of 2, and a height of 10.

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Answer

Write the formula to find the volume of a cone.

$V=\frac{1}{3}$\pi $r^2h$$

The diameter is 2, which means the radius is 1. Substitute the known dimensions.

$V=\frac{1}{3}$\pi $(1)^2$(10)$

The answer is: $$\frac{10}{3}$\pi$

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Question

A cone has height 240 centimeters; its base has radius 80 centimeters. Give its volume in cubic meters.

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Answer

Convert both dimensions from centimeters to meters by dividing by 100:

Height: 240 centimeters = $240 \div 100 = 2.4$ meters.

Radius: 80 centimeters = $80 \div 100 = 0.8$ meters.

Substitute in the volume formula:

$V = $\frac{1}{3}$ \pi $r^{2}$h$

$V = $\frac{1}{3}$ \cdot \pi \cdot 0.8 ^{2} \cdot 2.4$

$V = $\frac{1}{3}$\cdot 0.8 \cdot 0.8 \cdot 2.4 \cdot \pi$

$V =0.512 \pi$

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Question

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

Tap to reveal answer

Answer

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = $\frac{18}{12}$ = $\frac{3}{2}$$ feet

Radius: 4 inches = $4 \div 12 = $\frac{4}{12}$ = $\frac{1}{3}$$

The volume of a cone is given by the formula

$V = $\frac{1}{3}$ \pi $r^{2}$ h$

Substitute $r = $\frac{1}{3}$, h = $\frac{3}{2}$$ :

$V = $\frac{1}{3}$\cdot \left ( $\frac{1}{3}$ \right ) ^{2}\cdot $\frac{3}{2}$ \cdot \pi$

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

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

$V = $\frac{1}{18}$ \pi$

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Question

Give the volume of a cone whose height is 10 inches and whose base is a circle with circumference $6 \pi$ inches.

Tap to reveal answer

Answer

A circle with circumference $6 \pi$ inches has as its radius

$r = $\frac{C }{2\pi }$=\frac{6\pi }{2\pi }$ = 3$ inches.

The area of the base is therefore

$B = \pi $r^{2}$ = \pi \cdot $3^{2}$ = 9 \pi$ square inches.

To find the volume of the cone, substitute $B = 9 \pi , h = 10$ in the formula for the volume of a cone:

$V = $\frac{1}{3}$ Bh = $\frac{1}{3}$ \cdot 9 \pi \cdot 10 = 30 \pi$ cubic inches

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Question

The height of a cone and the radius of its base are equal. The circumference of the base is $10 \pi$ inches. Give its volume.

Tap to reveal answer

Answer

A circle with circumference $10 \pi$ inches has as its radius

$r = $\frac{C }{2\pi }$=\frac{10\pi }{2\pi }$ = 5$ inches.

The height is also inches, so substitute in the volume formula for a cone:

$V = $\frac{1}{3}$ \pi r ^{2}h = $\frac{1}{3}$ \pi \cdot 5 ^{2} \cdot 5 = $\frac{125}{3}$ \pi$ cubic inches

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Question

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

Tap to reveal answer

Answer

You are an architect designing a cone shaped structure. If the structure will be 30 ft tall and 10 feet wide at the base, what will the volume of the structure be?

Begin by using the formula for volume of a cone:

$V=\pi $r^2$$\frac{h}{3}$$

Now, we simply need to plug in our knowns.

We know the height is 30 ft.

We know that the diameter is 10ft, however, we need the radius.

Divide 10 by 2 to get a radius of 5 ft.

Now, let's go....

$V=\pi $(5ft)^2$$$\frac{30ft}{3}$=\pi25ft^2$*10ft=250 \pi $ft^3$$

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Question

Find the volume of a cone with the following measurements:

diameter = 12in

height = 6in

Tap to reveal answer

Answer

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

$V = \pi $r^2$ $\frac{h}{3}$$

where r is the radius and h is the height of the cone.

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

We know the height of the cone is 6in.

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

$V = \pi \cdot $(6$\text{in}$)^2$ \cdot $\frac{6\text{in}$}{3}$

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

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

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

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Question

Find the volume of a cone with the following measurements:

height: 12in

diameter: 6in

Tap to reveal answer

Answer

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

$V = \pi $r^2$ $\frac{h}{3}$$

where r is the radius and h is the height of the cone.

Now, we know the height of the cone is 12in. We also know the diameter of the cone is 6in. We know the diameter is two times the radius. Therefore, the radius is 3in.

So, we get

$V = \pi \cdot $(3$\text{in}$)^2$ \cdot $\frac{12\text{in}$}{3}$

$V = \pi \cdot $9$\text{in}$^2$ \cdot 4$\text{in}$$

$V = \pi \cdot $36$\text{in}$^3$$

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

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Question

Find the volume of a cone with the following measurements:

Diameter: 14in
Height: 9in

Tap to reveal answer

Answer

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

$V = \pi $r^2$ $\frac{h}{3}$$

where r is the radius and h is the height of the cone.

Now, we know the diameter of the cone is 14in. We also know the diameter is two times the radius. Therefore, the radius is 7in.

We know the height of the cone is 9in.

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

$V = \pi \cdot $(7$\text{in}$)^2$ \cdot $\frac{9\text{in}$}{3}$

$V = \pi \cdot $49$\text{in}$^2$ \cdot 3$\text{in}$$

$V = \pi \cdot $147$\text{in}$^3$$

$V = 147\pi $\text{ $in}$^3$$

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Question

Find the volume of a cone with a base diameter of 2, and a height of 10.

Tap to reveal answer

Answer

Write the formula to find the volume of a cone.

$V=\frac{1}{3}$\pi $r^2h$$

The diameter is 2, which means the radius is 1. Substitute the known dimensions.

$V=\frac{1}{3}$\pi $(1)^2$(10)$

The answer is: $$\frac{10}{3}$\pi$

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Question

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

Tap to reveal answer

Answer

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = $\frac{18}{12}$ = $\frac{3}{2}$$ feet

Radius: 4 inches = $4 \div 12 = $\frac{4}{12}$ = $\frac{1}{3}$$

The volume of a cone is given by the formula

$V = $\frac{1}{3}$ \pi $r^{2}$ h$

Substitute $r = $\frac{1}{3}$, h = $\frac{3}{2}$$ :

$V = $\frac{1}{3}$\cdot \left ( $\frac{1}{3}$ \right ) ^{2}\cdot $\frac{3}{2}$ \cdot \pi$

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

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

$V = $\frac{1}{18}$ \pi$

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Question

A cone has height 240 centimeters; its base has radius 80 centimeters. Give its volume in cubic meters.

Tap to reveal answer

Answer

Convert both dimensions from centimeters to meters by dividing by 100:

Height: 240 centimeters = $240 \div 100 = 2.4$ meters.

Radius: 80 centimeters = $80 \div 100 = 0.8$ meters.

Substitute in the volume formula:

$V = $\frac{1}{3}$ \pi $r^{2}$h$

$V = $\frac{1}{3}$ \cdot \pi \cdot 0.8 ^{2} \cdot 2.4$

$V = $\frac{1}{3}$\cdot 0.8 \cdot 0.8 \cdot 2.4 \cdot \pi$

$V =0.512 \pi$

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