Chapter 5.5, Problem 3E

Reminder Round all the answers to two decimal places unless otherwise indicated.

Cox’s Formula Assume that a long horizontal pipe connects the bottom of a reservoir with a drainage area. Cox’s formula provides a way of determining the velocity $v$ of the water flowing through the pipe:

$\frac{Hd}{L}=\frac{4v^2+5v-2}{1200}$.

Here $H$ is the depth of the reservoir in feet, $d$ is the pipe diameter in inches, $L$ is the length of the pipe in feet, and the velocity $v$ of the water is in feet per second. (See Figure $5.71$.)

a. Graph the quadratic function $4v^2+5v-2$, using a horizontal span from $0$ to $10$.

b. Judging on the basis of Cox’s formula, is it possible to have a velocity of $0.25$ foot per second?

c. Find the velocity of the water in the pipe if the pipe’s diameter is $4$ inches, its length is $1000$ feet, and the reservoir is $50$ feet deep.

d. If the water velocity is too high, there will be erosion problems. Assuming that the pipe length is $1000$ feet and the reservoir is $50$ feet deep, determine the largest pipe diameter that will ensure that the water velocity does not exceed $10$ feet per second.