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ACT Math : Cones

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Example Question #141 : Solid Geometry

Some conical party hats are being decorated with glitter paper. If the base of the hats is inches across and the slant height is inches, how many square inches of glitter paper are needed to decorate one hat?

Possible Answers:

$57\pi\textup{ in}^2$

$33\pi\textup{ in}^2$

$10\pi\textup{ in}^2$

$48\pi \textup{ in}^2$

$14\pi\textup{ in}^2$

Correct answer:

$48\pi \textup{ in}^2$

Explanation:

The formula for the surface area of a cone is

$\pi r^2 + \pi rl$ , where is the slant height. Since we're only concerned with the slant height portion of the surface area formula, we can ignore the $\pi r^2$ portion of the equation.

Plug in known values to this part of the equation and solve.

$\pi rl = \pi (6)(8) = 48\pi$

So, each hat requires $48\pi \textup{ in}^2$ of glitter paper.

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Example Question #142 : Solid Geometry

The surface area of a cone equals $24\pi \textup{cm}^2.$ If the radius of the cone equals $3\textup{cm}$ , what is the height of the cone?

Possible Answers:

$3\textup{cm}$

$8\textup{cm}$

$6\textup{cm}$

$5\textup{cm}$

$4\textup{cm}$

Correct answer:

$4\textup{cm}$

Explanation:

To solve this question, you need to know the surface area of a cone equation: $\textup{surface area}=\pi r(r+\sqrt{h^2 +r^2}),$ where is the radius of the cone $(3\textup{cm})$ , and is the height of the cone. We must substitute all of the values that we know, and solve for the height of the cone to get the answer.

$24\textup{cm}^2=\pi *3\textup{cm}(3\textup{cm}+\sqrt{h^2 +(3\textup{cm})^2}),$ Therefore, the height of the cone is $4\textup{cm.}$

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

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?

Possible Answers:

$45\pi$

$120\pi$

$90\pi$

$60\pi$

$30\pi$

Correct answer:

$60\pi$

Explanation:

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

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

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

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

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

$=\pi*90-\pi*30$

$=\pi*60$

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

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?

Possible Answers:

$24\pi$

$48\pi$

$8\pi$

$12\pi$

$6\pi$

Correct answer:

$24\pi$

Explanation:

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 1.5 .

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

2. Subtract the smaller volume from the larger volume:

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

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Example Question #11 : Cones

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?

Possible Answers:

$288\pi$

$1176\pi$

$276\pi$

$252\pi$

$300\pi$

Correct answer:

$252\pi$

Explanation:

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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Example Question #1061 : Act Math

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.

Possible Answers:

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

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

$112\pi cm^3$

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

Doubling the radius
Doubling the height
Quadrupling the height

Possible Answers:

1 and 3

3 only

2 only

1 only

1, 2 and 3

Correct answer:

1 and 3

Explanation:

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 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 #1 : How To Find The Volume Of A Cone

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.

Possible Answers:

$24\pi$

$231.32\pi$

$8\pi$

$36\pi$

$72\pi$

Correct answer:

$24\pi$

Explanation:

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

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

We plug in our given radius and height to find:

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

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Example Question #11 : Cones

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:

$387667\pi mm^2$

$784333\pi mm^2$

$106667\pi mm^2$

$233333\pi mm^2$

$99999\pi mm^2$

Correct answer:

$233333\pi mm^2$

Explanation:

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

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

Possible Answers:

$6\pi$

$72\pi$

$36\pi$

$18\pi$

$72\pi$

Correct answer:

$6\pi$

Explanation:

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

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

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ACT Math : Cones

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #141 : Solid Geometry

Example Question #142 : Solid Geometry

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

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

Example Question #11 : Cones

Example Question #1061 : Act Math

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

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

Example Question #11 : Cones

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

All ACT Math Resources

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