How to find the volume of a cone - ACT Math

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Question

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Answer

Cylinder Volume = $\pir^{2}h$

Cone Volume = $\frac{1}{3}\pir^{2}h$

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pir^{2}h-\frac{1}{3}\pir^{2}h$

$=\pi3^{2}10-\frac{1}{3}\pi3^{2}10$

$=\pi90-\pi30$

$=\pi60$

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Question

You have two cones, one with a diameter of and a height of and another with a diameter of and a height of . If you fill the smaller cone with sand and dump that sand into the larger cone, how much empty space will be left in the larger cone?

Answer

1. Find the volume of each cone:

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

Cone 1:

Since the diameter is , the radius is .

$Volume=\frac{1}{3}\pi (3^{2})(10)=30\pi$

Cone 2:

Since the diameter is , the radius is .

$Volume=\frac{1}{3}\pi (1.5^{2})(8)=6\pi$

2. Subtract the smaller volume from the larger volume:

$30\pi-6\pi=24\pi$

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Question

You have an empty cone and a cylinder filled with water. The cone has a diameter of and a height of . The cylinder has a diameter of and a height of . If you dump the water from the cylinder into the cone until it is filled, what volume of water will remain in the cylinder?

Answer

1. Find the volumes of the cone and cylinder:

Cone:

Since the diameter is , the radius is .

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

$Volume=\frac{1}{3}\pi (4^{2})(9)=48\pi$

Cylinder:

Since the diameter is , the radius is .

$Volume=\pi r^{2}h$

$Volume=\pi (5^{2})(12)=300\pi$

2. Subtract the cone's volume from the cylinder's volume:

$300\pi-48\pi=252\pi$

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Question

What is the volume of a cone with a height of 7 cm and a radius of 4 cm? Leave your answer in terms of $\pi$ and as a fraction if need be.

Answer

To find the volume of a cone plug the radius and height into the formula for the volume of a cone.

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

$\=\frac{1}{3}\pi 4^2\cdot 7\ \=\frac{112}{3}\pi cm^3$

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Question

What is the volume of a cone with a radius of 3 mm and a height of 6 mm?

Answer

The formula for the volume of a cone is given by the equation:
$V = \frac{1}{3} \pi r^2h$ .

Pluggin in our values we get:

$V = \frac{1}{3} \pi 3^26$

$V=18\pi mm^3$

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Question

Which of the following will quadruple the volume of a cone?

Doubling the radius

Doubling the height

Quadrupling the height

Answer

Bearing in mind the volume formula for a cone:

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

Because the volume varies by the square of the radius, doubling the radius will quadruple the volume (since .) Because the volume also varies linearly by the height, quadrupling the height will quadruple the volume.

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Question

What is the volume of a cone with a radius of and a height of ? Leave your answer in terms of $\pi$ , reduce all fractions.

Answer

To find the volume of a cone with radius , and height use the formula:

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

We plug in our given radius and height to find:

$V = \frac{1}{3}\pi * 6^2*2 = \frac{72}{3}\pi = 24\pi$

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Question

A conical paper cup is being used as a makeshift funnel for motor oil. If the cup is 100 millimeters deep at the center and has a radius of 70 millimeters, how many cubic millimeters of motor oil can it hold at one time? Round your final answer to the nearest integer.

Answer

The formula for the volume of a cone is:

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

With the information we have, we can simply plug values into this equation and solve.

$V = \frac{\pi r^2h}{3} = \frac{(100^2)(70)\pi}{3} \approx 233333\pi$

So our cone can hold approximately $233333\pi$ cubic millimeters of motor oil.

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Question

Find the volume of a cone whose diameter is and height is .

Answer

To find the volume of a cone, simply use the following formula. Thus,

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi3^22=\frac{1}{3}92\pi=6\pi$

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Question

Find the volume of a cone with radius and height of .

Answer

To solve simply use the formula for the voluma of a cone. Thus,

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi3^2*4=12\pi$

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