An agricultural sprinkler distributes water in a circular pattern of radius 115

Question

An agricultural sprinkler distributes water in a circular pattern of radius 115 ft. It supplies water to a depth of er feet per hour at a distance of r feet from the sprinkler. (Do not substitute numerical values; use variables only.) (a) If 0 < R 115, what is the total amount of water supplied per hour to the region inside the circle of radius R centered at 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 R.

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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