To show: The depth of the water decreases at a constant rate without regarding the shape of the bowl.
Answer to Problem 5P
The depth of the water decreases at a constant rate.
Explanation of Solution
Given information:
Water in an open bowl evaporates at a rate proportional to the area of the surface of the water.
Calculation:
The rate of decrease of the volume is proportional to the area of the surface.
Here, k is the positive constant and
Consider that the rate of change of depth with respect to time as
Apply Chain rule as shown below.
Substitute
Expression for the volume of water up to a depth x as shown below.
Here,
Differentiate both sides of the Equation with respect to x.
Substitute
Therefore, the depth of the water decreases at a constant rate without regarding the shape of the bowl. Hence, it is proved.
Chapter 6 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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