An agricultural sprinkler distributes water in a circular pattern of radius 110

Question

An agricultural sprinkler distributes water in a circular pattern of radius 110 ft. It supplies water to a depth of e^(-r) feet per hour at a distance of r feet from the sprinkler.

(b) Determine an expression for the average amount of water per hour per square foot supplied to the region inside the circle of radius

An agricultural sprinkler distributes water in a circular pattern of radius 110 ft. It supplies water to a depth of e^(-r) feet per hour at a distance of r feet from the sprinkler. (a) If 0

Explanation / Answer

(a) Using polar coordinates, this equals
?(? = 0 to 2?) ?(r = 0 to R) e^(-r) * (r dr d?)
= 2? ?(r = 0 to R) re^(-r) dr
= 2? [-re^(-r) - e^(-r)] {for r = 0 to R}
= 2? [1 - (R + 1) e^(-R)].

(b) Since the area of the region is ?R^2, the average value equals
(1/(?R^2)) * 2? [1 - (R + 1) e^(-R)] = (2/R^2) [1 - (R + 1) e^(-R)].

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