How to find the volume of a cone - SAT Math

Card 0 of 13

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)

Compare your answer with the correct one above

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

Compare your answer with the correct one above

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

Compare your answer with the correct one above

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.

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

Compare your answer with the correct one above

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

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

Compare your answer with the correct one above

Question

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

Answer

Write the formula to find the volume of a cone.

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

Substitute the known values and simplify.

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

Compare your answer with the correct one above

Question

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

Answer

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

$V=\frac{4}{3}\pi{r^2}h=\frac{4}{3}\pi4^23=4\pi*16=64\pi$

Compare your answer with the correct one above

Question

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

Answer

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

$V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi{3^2}5=\frac{1}{3}\pi95=15\pi$

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.

Compare your answer with the correct one above

Question

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?

Answer

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.

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

$27\pi = \pi r^3$

Compare your answer with the correct one above

Question

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

Answer

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 , but not .

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

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

Setting and , 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:

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

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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

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

and , so substitute and evaluate:

$V = \frac{1}{3} \pi \cdot 7^{2} \cdot 18$

$= \frac{1}{3} \pi \cdot 49 \cdot 18$

$=\left ( \frac{1}{3} \cdot 49 \cdot 18 \right ) \cdot \pi$

$= 294 \pi$

Compare your answer with the correct one above

Question

In terms of , express the volume of the provided right circular cone.

Answer

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

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

The height of the cone is shown to be equal to 20, so substituting accordingly:

$V = \frac{1}{3} \pi r^{2} \cdot 20$

$V = \frac{20}{3} \pi r^{2}$

Compare your answer with the correct one above

Question

In terms of , express the volume of the above right circular cone.

Answer

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

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

The slant height is shown in the diagram to be 24. By the Pythagorean Theorem,

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

Setting and solving for :

$h^{2}+ r ^{2} = 24^{2}$

$h^{2}+ r ^{2} = 576$

$h^{2}+ r ^{2} - r^{2} = 576 -r^{2}$

$h^{2} = 576 - r^{2}$

$h =\sqrt{ 576 - r^{2} }$

Substituting in the volume formula for :

$V = \frac{1}{3} \pi r^{2} \sqrt{ 576 - r^{2} }$ .

Compare your answer with the correct one above

Tap the card to reveal the answer