Firefighter Math: 3.2 Volume of Water in Hose

The volume of a hose allows an estimate of how much water can be delivered to the fire and is important in firefighting. The hose diameter is usually given in inches, with length in feet. The volume of a hose can be computed using the equation for volume of a cylinder in Section 3.1.

Example 1 - A 100-foot length of 1-inch diameter hose is charged with water. How many gallons of water are in that length of hose? 



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

Step 2. Identify the radius and height.
π = 3.14,     r = 1/2 × d = 1/2 × 1 in = 0.5 in,     h = 100 ft

Step 3. Convert all measurements to the same units. Convert the 100-foot hose length to inches.




Step 4. Set up the problem and solve. V = 3.14 × 0.5 in × 0.5 in × 1,200 in = 942 in³

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

The capacity of a 100-foot length of 1-inch diameter hose is 4 gallons.

ESTIMATING IN THE FIELD

Rounding makes numbers easier to use. Sometimes estimations or approximations are useful, such as for problem-solving in the field, especially when without a calculator. For example, from Table 3.1 we see that 100 feet of 1-1/2 inch hose holds 9.2 gallons of fluid. In the field, the conversion value can be rounded to 9 gallons when doing rough estimations/calculations. These rounded values are easier to remember.

Example 2 - Sherman is out on a fire. His crew has a trunk line of 6 lengths of 100-foot, 1 & 1/2 inch hose. He needs to estimate the volume of water in this trunk line. What is a good estimate?

Step 1. Find the appropriate conversion/estimation in Table 3.1. Commit these rounded values to memory.

The exact tables indicate that a single 100-foot length of 1-1/2-inch hose holds 9.2 gallons of water (Table 3.1). Use the rounded value of 9 gallons per 100-foot length of 1-1/2-inch hose.

Step 2. Set up the table. This may or may not need to be done on paper. Typical rough estimations involve only one multiplication or division step, and usually this math can be done in your head.



Sherman has used 54 gallons of water to charge the hose lay. 
Sherman now knows that he has about 54 gallons of water in his hose lay. The exact value would have been 6 × 9.2 = 55.2 gallons. 

Rounded numbers are easier to use, because multiplying whole numbers is simpler than multiplying decimals. In example 6, the number was rounded down from 9.2 to 9.0. If it would have been rounded up from 9.2 to 10.0, the solution would have been 6 × 10 = 60 gallons. By performing the calculations using whole numbers, both higher and lower than the actual value, a margin is created. This margin allows for an upper and lower limit. It is therefore safe to say that the actual value is between 54 gallons (6 × 9) and 60 gallons (6 × 10).

DETERMINING WEIGHTS AND VOLUMES OF WATER

Table 3.1 shows the volumes of water in specific hose lengths, along with the weight of 1 gallon of water. The weight of 1 gallon of water is 8.3 pounds (1 gallon = 8.3 pounds). With this conversion, the weight of water can be calculated in a certain length of hose or volume of water by multiplying by 8 pounds per gallon (rounded value).
 

Example 3 - The tank on a Model 62 Engine is filled with 500 gallons of water. How much weight does the water add to the weight of the engine?

Step 1. Find the appropriate estimation in Table 3.1. 1 gallon = 8 pounds (lb)

Step 2. Set up the cancellation table so all units will cancel, except the desired unit, pounds.



The water in the tank adds 4,000 pounds to the weight of the engine.
 

Example 4 - Two 100-foot lengths of 1 & 1/2-inch cotton-synthetic hose weigh about 54 pounds total when dry.
How much will the same hose weigh when fully charged?

Step 1. Find the appropriate estimation in Table 3.1. The volume capacity of one 1 & 1/2-inch ID × 100-foot hose length = 9 gallons. For two lengths of hose, there are two times 9 gallons. 



Step 2. Set up the cancellation table so all units will cancel, except the desired unit, pounds.



Step 3. Add the dry weight of the hose to the weight of the water. 54 lb + 144 lb = 198 lb

Two fully charged 1 & 1/2-inch ID × 100-foot hose lengths weigh 198 pounds.
 

Example 5 - An engine company is pumping a progressive hose lay with 1-inch laterals every 100 feet. At 800 feet up from the engine, the trunk line breaks. The firefighters replace it, but they forget to shut off the gated wye valve above the broken hose. As a result, they accidentally drain ten 100-foot lengths of 1-1/2-inch hose and ten 100-foot lengths of 1-inch hose. How much water above the break was lost due to this mistake?

Step 1. Find the appropriate estimation in Table 3.1 for the volume of water in both 1-inch and 1-1/2-inch lengths of hose.

Each 1-1/2-inch ID x 100-foot hose length holds 9 gallons.
Each 1-inch ID × 100-foot hose length holds 4 gallons.

Step 2. Set up the cancellation table so all units will cancel, except the desired unit, gallons, for each length of
hose.

Ten lengths of 1-1/2-inch ID x 100-foot hose



Ten lengths of 1-inch ID x 100-foot hose



Step 3. Add these together to find the amount of water lost. 90 gallons + 40 gallons = 130 gallons

The firefighters lost 130 gallons of water.