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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show that the depth of the water decreases at a constant rate, regardless of the shape of the bowl.

To determine

To show:

The depth of the water decreases at a constant rate regardless of the shape of the bowl.

Explanation

The rate of change of volume of water is given as

dVdt=-kAx   (1)

where k is some positive constant and Ax is the area of the surface when the water has depth x.

The rate of change of the depth of water with respect to time is  dxdt.

By the Chain rule,

dVdt=dVdxdxdt

From equation (1),

dVdt·dxdt=-kAx  (2)

The total volume of water up to

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