# PSAT Math : Cones

## Example Questions

### 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?

π(r2)(h)/(180w)

π(r2)(h)/(20w)

20w/(π(r2)(h))

π(r2)(h)/(60w)

180w/(π(r2)(h))

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)

### Example Question #2 : How To Find The Volume Of A Cone

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

1352π in3

338π in3

1014π in3

4394π in3

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)π13* 6 = 169 * 2π = 338π in3

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

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

142,296π in3

71,148π in3

11,858π in3

2964.5π in3

8893.5π in3

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

### Example Question #1 : Solid Geometry

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

Explanation:

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

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

### 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?

Explanation:

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 #1 : Cones

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

Explanation:

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:

meters

The slant height is 120 centimeters, which converts similarly to

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:

meters.

The volume formula can now be used:

cubic meters

### Example Question #2 : Cones

What is the volume of a cone with a height of  and base radius of ?

Explanation:

The fomula for the volume of a cone is

Because our radius of the base is , we know that .

.

### Example Question #163 : Solid Geometry

What is the surface area of a cone with a radius of 4 and a height of 3?

Explanation:

Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

### Example Question #162 : Solid Geometry

The lateral area is twice as big as the base area of a cone.  If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

81π

90π

27π

54π

81π

Explanation:

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r2)

rl = 2r2

l = 2r

From the diagram, we can see that r2 + h2 = l2.  Since h = 9 and l = 2r, some substitution yields

r2 + 92 = (2r)2

r2 + 81 = 4r2

81 = 3r2

27 = r2

B = π(r2) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

### Example Question #11 : Surface Area

A right cone has a radius of 4R and a height of 3R. What is the ratio of the total surface area of the cone to the surface area of just the base?

Explanation:

We need to find total surface area of the cone and the area of the base.

The area of the base of a cone is equal to the area of a circle. The formula for the area of a circle is given below:

, where r is the length of the radius.

In the case of this cone, the radius is equal to 4R, so we must replace r with 4R.

To find the total area of the cone, we need the area of the base and the lateral surface area of the cone. The lateral surface area (LA) of a cone is given by the following formula:

, where r is the radius and l is the slant height.

We know that r = 4R. What we need now is the slant height, which is the distance from the edge of the base of the cone to the tip.

In order to find the slant height, we need to construct a right triangle with the legs equal to the height and the radius of the cone. The slant height will be the hypotenuse of this triangle. We can use the Pythagorean Theorem to find an expression for l. According to the Pythagorean Theorem, the sum of the squares of the legs (which are 4R and 3R in this case) is equal to the square of the hypotenuse (which is the slant height). According to the Pythagorean Theorem, we can write the following equation:

Let's go back to the formula for the lateral surface area (LA).

To find the total surface area (TA), we must add the lateral area and the area of the base.

The problem requires us to find the ratio of the total surface area to the area of the base. This means we must find the following ratio:

We can cancel , which leaves us with 36/16.

Simplifying 36/16 gives 9/4.