How to find the volume of a cone - PSAT Math

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Question

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?

Answer

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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Question

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

Answer

The basic form for the volume of a cone is:

V = (1/3)πr_2_h

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

V = (1/3)π_132 6 = 169 * 2_π* = 338_π_ in3

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Question

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

Answer

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 = 2_πr_, or, for our values 77_π_ = 2_πr_. 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)πr_2_h

For our values this would be:

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

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Question

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

Answer

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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Question

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?

Answer

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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Question

Which of these answers comes closest to the volume of the above cone?

Answer

The radius and the height of a cone are required in order to find its volume.

The radius is 50 centimeters, which can be converted to meters by dividing by 100:

$50 \div 100 = 0.5$ meters

The slant height is 120 centimeters, which converts similarly to

$120 \div 100 = 1.2$ meters

To find the height, we need to use the Pythagorean Theorem with the radius 0.5 as one leg and the slant height 1.2 as the hypotenuse of a right triangle, and the height as the other leg:

$h = \sqrt{1.2^{2} -0.5^{2}} = \sqrt{1.44 -0.25} = \sqrt{1.19} \approx 1.09$ meters.

The volume formula can now be used:

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

$\approx \frac{3.14 \cdot 0.5^{2} \cdot 1.09}{3}$

$\approx \frac{3.14 \cdot 0.25 \cdot 1.09}{3}$

$\approx \frac{0.8557}{3}$

$\approx 0.29$ cubic meters

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Question

What is the volume of a cone with a height of $12\textup{\textup{ inches}}$ and base radius of $5\textup{ inches}$ ?

Answer

The fomula for the volume of a cone is

$\textup{where }b\textup{ is the area of the base, and }h\textup{ is the height}$

Because our radius of the base is $5\textup{ inches}$ , we know that $area=\pi r^2=25\pi$ .

$v=\left ( \frac{1}{3} \right )(12)(25\pi)=100\pi$ .

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