### All SAT Math Resources

## Example Questions

### Example Question #1 : Cones

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(r^{2})(h)/(20w)

π(r^{2})(h)/(180w)

20w/(π(r^{2})(h))

180w/(π(r^{2})(h))

π(r^{2})(h)/(60w)

**Correct answer:**

π(r^{2})(h)/(180w)

The volume of a cone is given by the formula V = (πr^{2})/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 = (πr^{2})/(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.

(πr^{2})/(3w) * (1/60) = π(r^{2})(h)/(180w)

### Example Question #1 : Cones

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

**Possible Answers:**

338*π* in^{3}

1014*π* in^{3}

4394*π* in^{3}

1352*π* in^{3}

None of the other answers

**Correct answer:**

338*π* in^{3}

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*13^{2 }* 6 = 169 * 2*π* = 338*π* in^{3}

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

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

**Possible Answers:**

11,858*π* in^{3}

71,148*π* in^{3}

8893.5*π* in^{3}

142,296*π* in^{3}

2964.5*π* in^{3}

**Correct answer:**

11,858*π* in^{3}

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* * 38.5^{2} * 24 = 8 * 1482.25*π* = 11,858π in^{3}

### Example Question #2 : Cones

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

**Possible Answers:**

**Correct answer:**

The general formula is given by , where = radius and = height.

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

### Example Question #2 : Solid Geometry

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:**

**Correct answer:**

First we will calculate the volume of the cone

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

We will then convert that into minutes

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

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

**Possible Answers:**

**Correct answer:**

Write the formula to find the volume of a cone.

Substitute the known values and simplify.

### 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:**

**Correct answer:**

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

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.

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

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

**Possible Answers:**

**Correct answer:**

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

### Example Question #7 : How To Find The Volume Of A Cone

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

**Possible Answers:**

**Correct answer:**

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

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

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

### Example Question #8 : How To Find The Volume Of A Cone

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

**Possible Answers:**

**Correct answer:**

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

.

We are given , but not .

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

Setting and , substitute and solve for :

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

Now, substitute for and and evaluate: