   Chapter 15.3, Problem 36E

Chapter
Section
Textbook Problem

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.(a) 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?(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.

(a)

To determine

To find: The total amount of water supplied to the region inside the circle of radius R, where 0<R100.

Explanation

Given:

The agricultural sprinkler distributes water in a circular pattern of radius 100 ft and supplies water to the depth of er.

Formula used:

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π, then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

Since the agricultural sprinkler distributes water in the circular shape, r varies from 0 to R and θ varies from 0 to 2π. It is given that depth of the water will be er. So, by the equation (1) the total amount of water supplied is,

DzdA=02π0Rer(r)drdθ=02π0Rrerdrdθ

Integrate the function with respect to r and θ by using the equation (2).

02π0Rrerdrdθ=02πdθ0Rrerdr

Apply the technique of integration by parts and obtain the required integral value.

Let u=r.

Then, dv=erdr

(b)

To determine

The average amount of water supplied to the region inside the circle of radius R, where 0<R100.

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