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SAT Math : How to find the volume of a cone

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

An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?

Possible Answers:

π(r²)(h)/(60w)

π(r²)(h)/(180w)

π(r²)(h)/(20w)

180w/(π(r²)(h))

20w/(π(r²)(h))

Correct answer:

π(r²)(h)/(180w)

Explanation:

The volume of a cone is given by the formula V = (πr²)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.

time in seconds = (πr²)/(3w)

In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.

(πr²)/(3w) * (1/60) = π(r²)(h)/(180w)

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Example Question #1392 : Concepts

A cone has a base radius of 13 in and a height of 6 in. What is its volume?

Possible Answers:

None of the other answers

1014π in³

4394π in³

1352π in³

338π in³

Correct answer:

338π in³

Explanation:

The basic form for the volume of a cone is:

V = (1/3)πr²h

For this simple problem, we merely need to plug in our values:

V = (1/3)π13²* 6 = 169 * 2π = 338π in³

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Example Question #872 : Sat Mathematics

A cone has a base circumference of 77π in and a height of 2 ft. What is its approximate volume?

Possible Answers:

8893.5π in³

11,858π in³

71,148π in³

2964.5π in³

142,296π in³

Correct answer:

11,858π in³

Explanation:

There are two things to be careful with here. First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2πr, or, for our values 77π = 2πr. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr²h

For our values this would be:

V = (1/3)π * 38.5² * 24 = 8 * 1482.25π = 11,858π in³

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Example Question #3 : Solid Geometry

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

Possible Answers:

$120\pi \ cm^{3}$

$180\pi \ cm^{3}$

$15\pi \ cm^{3}$

$45\pi \ cm^{3}$

$60\pi \ cm^{3}$

Correct answer:

$15\pi \ cm^{3}$

Explanation:

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ , where = radius and = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$

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

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

Possible Answers:

$60\ \textup{minutes}$

$3\ \textup{minutes}\ 20\ \textup{seconds}$

$800\ \textup{minutes}$

$600\ \textup{minutes}$

$41\ \textup{minutes}\ 53\ \textup{seconds}$

Correct answer:

$41\ \textup{minutes}\ 53\ \textup{seconds}$

Explanation:

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

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

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

Possible Answers:

$9000\pi$

$900\pi$

$3000\pi$

$300\pi$

$6000\pi$

Correct answer:

$3000\pi$

Explanation:

Write the formula to find the volume of a cone.

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

Substitute the known values and simplify.

$V=\frac{1}{3}\pi (10)^2 (90)= \frac{1}{3}\pi (100)(90)= \pi (100)(30)=3000\pi$

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

Find the volume of a cone with radius 3 and height 5.

Possible Answers:

$60\pi$

$45\pi$

$30\pi$

$15\pi$

Correct answer:

$15\pi$

Explanation:

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

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi*{3^2}*5=\frac{1}{3}\pi*9*5=15\pi$

To remember the formula for volume of a cone, it helps to break it up into it's base and height. The base is a circle and the height is just h. Now, just multiplying those two together would give you the formula of a cylinder (see problem 3 in this set). So, our formula is going to have to be just a portion of that. Similarly to volume of a pyramid, that fraction is one third.

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Example Question #871 : Sat Mathematics

Find the area of a cone whose radius is 4 and height is 3.

Possible Answers:

$48\pi$

$16\pi$

$24\pi$

$64\pi$

Correct answer:

$64\pi$

Explanation:

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

$V=\frac{4}{3}\pi{r^2}h=\frac{4}{3}*\pi*4^2*3=4*\pi*16=64\pi$

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

The volume of a right circular cone is $9\pi$ . If the cone's height is equal to its radius, what is the radius of the cone?

Possible Answers:

$\sqrt{27}$

$3\pi$

Correct answer:

Explanation:

The volume of a right circular cone with radius and height is given by:

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

Since the height of this cone is equal to its radius, we can say:

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

Now, we can substitute our given volume into the equation and solve for our radius.

$9\pi = \frac{1}{3}\pi r^3$

$27\pi = \pi r^3$

27 = r^3

r = 3

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

Cone

The above is a right circular cone. Give its volume.

Possible Answers:

$175 \pi$

$49 \pi+35 \pi\sqrt{11}$

$294 \pi$

$\frac{245}{3} \pi \sqrt{11}$

$\frac{1,925}{3} \pi$

Correct answer:

$\frac{245}{3} \pi \sqrt{11}$

Explanation:

The volume of a right circular cone can be calculated from its height and the radius of its base using the formula

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

We are given r = 7 , but not .

, , and the slant height of a right circular cone are related by the Pythagorean Theorem:

$h^{2}+ r^{2}= l^{2}$

Setting l = 18 and r = 7 , substitute and solve for :

$h^{2}+ 7^{2}= 18^{2}$

$h^{2}+49= 324$

$h^{2}+49 - 49 = 324 - 49$

$h^{2} =275$

Taking the square root of both sides and simplifying the radical:

$h =\sqrt{275} = \sqrt{25} \cdot \sqrt{11} = 5 \sqrt{11}$

Now, substitute for and and evaluate:

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

$V = \frac{1}{3} \pi \cdot 7^{2} \cdot 5 \sqrt{11}$

$V = \frac{1}{3} \pi \cdot 49 \cdot 5 \sqrt{11}$

$V = \frac{245}{3} \pi \sqrt{11}$

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SAT Math : How to find the volume of a cone

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

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

Example Question #1392 : Concepts

Example Question #872 : Sat Mathematics

Example Question #3 : 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 #1 : How To Find The Volume Of A Cone

Example Question #871 : Sat Mathematics

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

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

All SAT Math Resources

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