3.1 Volume or Capacity

Volume is used to indicate the capacity of a tank or container. It is used by firefighters to answer questions like "How much water is left in the tank?" and "At 15 gallons per minute (gpm), how many more minutes before the tank is empty?"

### Volume of a Rectangular Object

The volume of a rectangular container is determined by multiplying the length (l) by the width (w) by the height (h).

Volume = length × width × height
V = l × w × h

Example 1 - Determine the volume of the tank in gallons.

Step 1. Use the equation for determining the volume of a rectangle. V = l × w × h

Step 2. Identify the length, width, and height. l = 5 ft, w = 6 ft, h = 8 ft

Step 3. Set up the problem and solve. V = 5ft × 6ft × 8ft = (5 × 6 × 8) (ft × ft × ft) = 240 ft3

Step 4. Determine the appropriate conversion factor. 1 cubic foot = 7.4805 gallons

Step 5. Set up the cancellation table so all units will cancel except gallons (see Section 2.1).

The volume of the tank of water is 240 cubic feet or 1,795 gallons.

Example 2 - The water tank on a newly designed engine is 34 inches wide, 5 feet high, and 12 feet long. What is the capacity of the water tank in cubic feet? In gallons?

Step 1. Use the equation for determining the volume of a rectangle. V = l × w × h

Step 2. Identify the length, width, and height. l = 12 ft, w = 34 in, h = 5 ft

Step 3. Convert all the measurements to the same units, feet.

Step 4. Set up the problem, and solve for volume. V = 2.83 ft × 5 ft × 12 ft = 170 ft3

Step 5. Set up the cancellation table so all units will cancel, except the desired unit, gallons.

The volume of the tank is 170 cubic feet or 1272 gallons.

### Volume of a cylinder

The volume of a cylinder is found by multiplying the area of the base times the height, h. The base of a cylinder is a circle, A = × r2, where = 3.14.

Example 3 - A cylindrical tank of foam concentrate is 5 feet tall. The tank diameter is 2.5 feet. What is the capacity, in gallons, of the tank?

Step 1. Use the equation for determining the volume of a cylinder.
V = r2 × h

Step 2. Draw a sketch. Label the height and diameter.

= 3.14, r = 1/2 d = 1/2 × 2.5 ft = 1.25 ft, h = 5 ft

Step 3. Set up the problem and solve. V = 3.14 × (1.25 ft)2 × 5 ft = 24.5 ft3

Step 4. Set up the cancellation table so all units will cancel, except the desired unit, gallons.

The tank capacity is 183.3 gallons.

### Practice

1. Carla is filling the cylindrical tank 8 feet in diameter and 10 feet deep with a pump. She’s pumping at a rate of 13 gallons per minute (gpm). How long will it take to fill the tank? Proceed through the questions below, which correspond to the steps necessary to solve the problem. After each step, click the next question button until you have reached the end of the exercise and have a value, in hours, for the time needed to fill the tank.

Part 1. The first step in solving this problem is to calculate the volume of the tank. What equation should you use to calculate the volume?

The correct answer is: (a) the volume of a cylindrical tank is the area ( times the radius squared) times the height.

Part 2. Set up the calculation, filling in the appropriate values below.

× ×
× r2 × h

= 3.14, r=1/2 × 8 = 4 ft, r2=42=16, h=10 ft.

Part 3. Use a calculator to solve for the volume. The volume of the tank is:

1004.8 ft³

223.11 ft³

Answer: (b), the volume of the tank is 502.04 ft³.

Part 4. Calculate the number of minutes it will take Carla to fill the tank. First, convert the volume of the tank from ft³ to gallons using the conversion table

Fill in the appropriate values below.

 × ( / 1 ft3 ) = ft3 × gallons/ft3 = gallons

From the table, 1 ft3 equals 7.48 gallons. Setting up the cancellation table and multiplying, we find that 502.04 ft3 × 7.48 gallons/1 ft3 = 3755 gallons

Part 5. The length of time needed to fill the tank is the volume of the tank divided by the rate of pumping. Fill in the blanks below with the appropriate values.

 / = gallons / gallons per min = minutes to fill tank

Dividing 3755 gallons by 13 gallons/min gives an answer of 289 minutes to fill the tank.

Part 6. Simplify the answer in minutes to the appropriate number of hours. Fill in the blanks with the appropriate values to cancel the minutes units and get an answer in hours.

 × / = minutes × factor for units cancellation = time in hrs

To convert hours to minutes, set up the cancellation table to multiply by 1 hr/60 min. 3755 × 1/60 = 3755 ÷ 60 = 4.81 hrs. It will take Carla nearly 5 hours to fill the tank.