Chapter 14, Problem 16PS

SprinklerConsider a circular lawn with a radius of 10 feet, as shown in the figure. Assume that a sprinkler distributes water in radial fashion according to the formula

$f(r)=\frac{r}{16}-\frac{r^2}{160}$

(measured in cubic feet of water per hour per square foot of lawn), where $r$ is the distance in feet from the sprinkler. Find the amount of water that is distributed in 1 hour in the following two annular regions.

$A=\left\{(r,\theta ):4\le r\le 5,0\le \theta \le 2\pi \right\}$

$B=\left\{(r,\theta ):9\le r\le 10,0\le \theta \le 2\pi \right\}$

Is the distribution of water uniform? Determine the amount of water the entire lawn receives in 1 hour.