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ACT Math : Solid Geometry

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Example Questions

Example Question #151 : Solid Geometry

You have an empty cone and a cylinder filled with water. The cone has a diameter of $\displaystyle 8$ and a height of $\displaystyle 9$ . The cylinder has a diameter of $\displaystyle 10$ and a height of $\displaystyle 12$ . If you dump the water from the cylinder into the cone until it is filled, what volume of water will remain in the cylinder?

Possible Answers:

$252\pi$ $\displaystyle 252\pi$

$1176\pi$ $\displaystyle 1176\pi$

$288\pi$ $\displaystyle 288\pi$

$300\pi$ $\displaystyle 300\pi$

$276\pi$ $\displaystyle 276\pi$

Correct answer:

$252\pi$ $\displaystyle 252\pi$

Explanation:

1. Find the volumes of the cone and cylinder:

Cone:

Since the diameter is $\displaystyle 8$ , the radius is $\displaystyle 4$ .

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

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

Cylinder:

Since the diameter is $\displaystyle 10$ , the radius is $\displaystyle 5$ .

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

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

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

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

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Example Question #3 : How To Find The Volume Of A Cone

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 $\pi$ $\displaystyle \pi$ and as a fraction if need be.

Possible Answers:

$\frac{448}{3}\pi cm^3$ $\displaystyle \frac{448}{3}\pi cm^3$

$\frac{56}{3} \pi cm^3$ $\displaystyle \frac{56}{3} \pi cm^3$

$\frac{112}{3}\pi cm^3$ $\displaystyle \frac{112}{3}\pi cm^3$

$112\pi cm^3$ $\displaystyle 112\pi cm^3$

$\frac{196}{3}\pi cm^3$ $\displaystyle \frac{196}{3}\pi cm^3$

Correct answer:

$\frac{112}{3}\pi cm^3$ $\displaystyle \frac{112}{3}\pi cm^3$

Explanation:

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

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

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

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Example Question #152 : Solid Geometry

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

Doubling the radius
Doubling the height
Quadrupling the height

Possible Answers:

1, 2 and 3

1 and 3

1 only

3 only

2 only

Correct answer:

1 and 3

Explanation:

Bearing in mind the volume formula for a cone:

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

Because the volume varies by the square of the radius, doubling the radius will quadruple the volume (since 2^2 = 4 $\displaystyle 2^2 = 4$ .) Because the volume also varies linearly by the height, quadrupling the height will quadruple the volume.

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Example Question #3 : How To Find The Volume Of A Cone

What is the volume of a cone with a radius of $\displaystyle 6$ and a height of $\displaystyle 2$ ? Leave your answer in terms of $\pi$ $\displaystyle \pi$ , reduce all fractions.

Possible Answers:

$72\pi$ $\displaystyle 72\pi$

$8\pi$ $\displaystyle 8\pi$

$231.32\pi$ $\displaystyle 231.32\pi$

$24\pi$ $\displaystyle 24\pi$

$36\pi$ $\displaystyle 36\pi$

Correct answer:

$24\pi$ $\displaystyle 24\pi$

Explanation:

To find the volume of a cone with radius $\displaystyle r$ , and height $\displaystyle h$ use the formula:

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

We plug in our given radius and height to find:

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

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Example Question #2 : How To Find The Volume Of A Cone

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.

Possible Answers:

$233333\pi mm^2$ $\displaystyle 233333\pi mm^2$

$106667\pi mm^2$ $\displaystyle 106667\pi mm^2$

$99999\pi mm^2$ $\displaystyle 99999\pi mm^2$

$784333\pi mm^2$ $\displaystyle 784333\pi mm^2$

$387667\pi mm^2$ $\displaystyle 387667\pi mm^2$

Correct answer:

$233333\pi mm^2$ $\displaystyle 233333\pi mm^2$

Explanation:

The formula for the volume of a cone is:

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

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

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

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

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Example Question #1 : How To Find The Volume Of A Cone

Find the volume of a cone whose diameter is $\displaystyle 6$ and height is $\displaystyle 2$ .

Possible Answers:

$72\pi$ $\displaystyle 72\pi$

$18\pi$ $\displaystyle 18\pi$

$36\pi$ $\displaystyle 36\pi$

$6\pi$ $\displaystyle 6\pi$

$72\pi$ $\displaystyle 72\pi$

Correct answer:

$6\pi$ $\displaystyle 6\pi$

Explanation:

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

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}*\pi*3^2*2=\frac{1}{3}*9*2*\pi=6\pi$ $\displaystyle V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}*\pi*3^2*2=\frac{1}{3}*9*2*\pi=6\pi$

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Example Question #3 : How To Find The Volume Of A Cone

Find the volume of a cone with radius $\displaystyle 3$ and height of $\displaystyle 4$ .

Possible Answers:

$12\pi$ $\displaystyle 12\pi$

$108\pi$ $\displaystyle 108\pi$

$36\pi$ $\displaystyle 36\pi$

$28\pi$ $\displaystyle 28\pi$

Correct answer:

$12\pi$ $\displaystyle 12\pi$

Explanation:

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

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

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Example Question #4 : How To Find The Volume Of A Cone

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

Possible Answers:

$108 \pi\textup{ mm}^3$ $\displaystyle 108 \pi\textup{ mm}^3$

$6 \pi\textup{ mm}^3$ $\displaystyle 6 \pi\textup{ mm}^3$

$36 \pi\textup{ mm}^3$ $\displaystyle 36 \pi\textup{ mm}^3$

$54\pi\textup{ mm}^3$ $\displaystyle 54\pi\textup{ mm}^3$

$18\pi\textup{ mm}^3$ $\displaystyle 18\pi\textup{ mm}^3$

Correct answer:

$18\pi\textup{ mm}^3$ $\displaystyle 18\pi\textup{ mm}^3$

Explanation:

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

Pluggin in our values we get:

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

$V=18\pi mm^3$ $\displaystyle V=18\pi mm^3$

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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #151 : Solid Geometry

Example Question #3 : How To Find The Volume Of A Cone

Example Question #152 : Solid Geometry

Example Question #3 : How To Find The Volume Of A Cone

Example Question #2 : How To Find The Volume Of A Cone

Example Question #1 : How To Find The Volume Of A Cone

Example Question #3 : How To Find The Volume Of A Cone

Example Question #4 : How To Find The Volume Of A Cone

All ACT Math Resources

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