Question

An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of e^-r feet per hour at a distance of r feet from the sprinkler. If 0 < R ≤ 100, what is the total amount of water supplied per hour to the region inside the circle of radius R centered at the sprinkler?

Answer 1

A circle of radius
`R` ft has area
`\pi R^2` sq ft. For any fixed
`R`, water will reach a depth of
`e^(-R)` ft. You can think of the total volume of water supplied within a radius
`R` of the sprinkler as the volume of the "cylinder" with "height" given by
`e^(-r)` for some
`0<r\le R`.

This volume is

`\displaystyle\int_0^(2\pi)\int_0^Rre^(-r)\,\mathrm dr\,\mathrm d\theta=2\pi(1-e^(-R)(R+1))`

cu ft.