   Chapter 15.3, Problem 35E

Chapter
Section
Textbook Problem

A swimming pool is circular with a 40-ft diameter. The depth is constant along east-west lines and increases linearly from 2 ft at the south end to 7 ft at the north end. Find the volume of water in the pool.

To determine

To find: The volume of water in the swimming pool.

Explanation

Given:

The swimming pool is in circular shape of diameter 40 ft.

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 swimming pool is in the circular shape, r varies from 0 to 20 and θ varies from 0 to 2π. It is given that the depth of the pool increases linearly from South to North. Thus, assume the function as h(y)=ay+b and from the given conditions it is observed that,

h(20)=2h(20)=7

Substitute the values in the equation h(y)=ay+b and obtain the values of a and b.

Case (i): h(20)=2

h(y)=ay+bh(20)=a(20)+b

2=20a+b (3)

Case (ii): h(20)=7

h(y)=ay+bh(20)=a(20)+b

7=20a+b (4)

Solve the equations (3) and (4). That is, add both the equations yields,

9=2bb=92

And, substitute the value of b in the equation (3),

7=20a+92792=20a52=20aa=18

Therefore, h(y)=18y+92 and by the equation (1) the volume of the water in the swimming pool becomes,

DzdA=02π020(18rsinθ+92)(r)drdθ=02π020(18r2sinθ+92r)drdθ=02π02018r2sinθdrdθ+02π0209r2drdθ

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

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