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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/ min. How fast is the height of the water increasing?

To determine

To find: The rate of change of the water level of the cylindrical tank whose radius is 5 m and being filled with water at a rate of 3m3/min.

Explanation

Given:

The radius of the cylindrical tank is 5 m.

The rate at which the cylindrical tank being filled with water is 3m3/min.

Formula used:

(1) Chain rule: dydx=dydududx.

(2) Volume of cylinder with radius r and height h: V=πr2h.

Calculation:

Let V be the volume of the cylindrical tank, r be the radius of the cylindrical tank and h be the height of the water level in a cylindrical tank.

The volume and height of the water in a cylindrical tank increases when the time t increases.

That is, the volume and height of the water level depends on the time variable t.

Thus, the volume V and the height h is a function of the time variable t.

Since the cylindrical tank being filled with water at a rate of 3m3/min, dVdt=3m3/min.

Obtain the derivative dhdt when the radius is 5 m.

First, substitute 5 for r in the volume of cylinder.

V=π(5)2h=π(25)h=25πh

Differentiate V with respect to the time variable t,

dVdt=ddt<

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