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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis.

(a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to pump the water out of the tank.

(b) After 4000 ft–lb of work has been done, what is the depth of the water remaining in the tank?

Chapter 5.R, Problem 29E, A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is,

To determine

(a)

To find:

The work required to pump the water out of the tank.

Explanation

1) Concept:

The work is calculated by using the formula W=abf(x)dx

2) Given:

The height of the paraboloid is 4 ft.

The radius of the paraboloid is 4 ft.

3) Calculations:

The vertex of the parabola is at origin and passing through point 4,4.

Therefore, the equation of the parabola is y=ax2.

Substitute values of (x, y) from the given point

4=a·42=16a

Solving for a,

a=14

Hence,y=14x2

That is,x2=4y

Therefore, x=2y.

Each circular disk has radius 2y and is moved 4-y ft.

And water weighs 62.5 lb/ft3

Therefore, the work required to pump the water out of the tank is

W=0462

To determine

(b)

To find:

The depth of water remaining in the tank after 4000 ft·lb of work has been done.

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