Chapter 2.1, Problem 30E

Discharge from a Fire Hose The discharge from a fire hose depends on the diameter of the nozzle. Nozzle diameters are normally in multiples of $\frac{1}{8}$ inch. Sometimes it is important to replace several hoses with a single hose of equivalent discharge capacity. Hoses with nozzle diameters $d_1,d_2,\mathrm{...},d_n$ have the same discharge capacity as a single hose with nozzle diameter D, where

$D=\sqrt{d_1^2+d_2^2+\mathrm{...}+d_n^2.}$

a. A nozzle of what diameter has the same discharge capacity as three combined nozzles of diameters $1\frac{1}{8}$ inches, $1\frac{5}{8}$ inches, and $1\frac{7}{8}$ inches? You should report your answer as an available nozzle size, that is, in multiples of $\frac{1}{8}.$

b. We have two 1-inch nozzles and wish to use third so that the combined discharge capacity of the three nozzles is the same as the discharge capacity of a $2\frac{1}{4}$ inch nozzle. What should be the diameter of the third nozzle?

c. If we wish to use n hoses, each with nozzle size d, in order to have the combined discharge capacity of a single hose with nozzle size D, then we must use

$n={\left(\frac{D}{d}\right)}^2$ nozzles.

How many half-inch nozzles are needed to attain the discharge capacity of a 2-inch nozzle?

d. We want to replace a nozzle of diameter $2\frac{1}{4}$ inches with 4 hones, each of the same nozzle diameter. What nozzle diameter for the 4 hoses will produce the same discharge capacity as the single hose?