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Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636
BuyFind

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636

Solutions

Chapter
Section
Chapter 5.4, Problem 22E
Textbook Problem

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.

22.    A spherical water tank, 24 ft in diameter, sits atop a 60 ft tower. The tank is filled by a hose attached to the bottom of the sphere. If a 1.5 horsepower pump is used to deliver water up to the tank, how long will it take to fill the tank? (One horsepower = 550 ft-lb of work per second.)

Expert Solution
To determine

To express: The work done to fill the water in the water tank as Riemann sum.

To calculate: The time required to fill the tank.

Explanation of Solution

Given:

The diameter d of the spherical water tank is 24 ft.

The tank sits at a top of a tower of 60 ft height.

The pump used to deliver water up to the tank is 1.5 horsepower.

The relation for conversion of power from horse power to ft-lb as follows:

1horsepower=550ft-lbsec

Write the expression to find the work done W to pump the water from a to b, when a force f(x) is applied on it.

W=limni=1nf(xi)Δx=abf(x)dx (1)

Calculation:

Assume the value of acceleration due to gravity g as 9.8m/s2.

Consider the centre point O of the water tank as the origin, introduce a vertical coordinate line extending downwards as shown in Figure 1.

The tank extends from 12ft to 12 ft.

Divide the water in the tank into n slices.

Consider the thin slice of water at a distance x ft below the centre of the water tank as shown in Figure 2.

Refer to Figure 2

Consider the radius of the horizontal slice as rs.

Calculate the radius of the horizontal slice as shown below:

122=x2+rs2rs2=122x2rs=122x2

Calculate the area of the horizontal slice as shown below:

Area=πrs2=π(122x2)2=π(122x2)

Consider the thickness of the horizontal slice as Δx.

Calculate the volume of the horizontal slice as shown below:

Volume=Area×thickness=πrs2Δx=π(122x2)Δx

Calculate the weight of the water in the horizontal slice.

Consider the weight density of water as 62.5lb/ft3.

Weight=Volume×WeightDensity=π(122x2)Δx×62.5=62.5π(122x2)Δx

Calculate the work done to pump the water to the level of horizontal slice.

The water has to be pumped from ground level (x=72) to the level of horizontal slice (x=x).

Thus, the distance the water has to be pumped is (72x).

Calculate the work done to lift the water using the relation:

Wi=Force×distance=Weight of water×distance=62.5π(122x2)Δx×(72x)=62.5π(122x2)(72x)Δx

Calculate the total work done W to pump entire water in the tank.

Taking Summation of work done by n slices.

W=i=1n62.5π(122x2)(72x)Δx (2)

Thus, the work done to pump the water into the tank is expressed as Riemann sum as i=1n62

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Chapter 5 Solutions

Single Variable Calculus
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