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CalculusSingle Variable CalculusShow 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.)BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

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.

**Given:**

The diameter

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:

Write the expression to find the work done

**Calculation:**

Assume the value of acceleration due to gravity

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

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

Calculate the radius of the horizontal slice as shown below:

Calculate the area of the horizontal slice as shown below:

Consider the thickness of the horizontal slice as

Calculate the volume of the horizontal slice as shown below:

Calculate the weight of the water in the horizontal slice.

Consider the weight density of water as

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

The water has to be pumped from ground level

Thus, the distance the water has to be pumped is

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

Calculate the total work done

Taking Summation of work done by

Thus, the work done to pump the water into the tank is expressed as Riemann sum as

Single Variable Calculus

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