Chapter 2.P, Problem 27P

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is
*r* is the radius and *l* is the slant height.) If we pour the liquid into the container at a rate of 2cm^{3}/min, then the height of the liquid decreases at a rate of 0.3 cm/min when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container?

To determine

**To find:** at what rate should we pour the liquid into the container

Explanation

**1) Concept:**

Differentiating the volume of cone and equating it with difference of what is being added and what is oozing out and solving further to maintain height to 10 cm

**2) Formula:**

i) Volume of cone:

ii) Surface area of cone:

**3) Given:**

i.

ii.

iii. When height is 10 cm,

iv. Liquid oozing out =

v. Liquid added =

4) **Calculation:**

By similar triangle,

The volume of inner cone is,

Differentiate with respect to t,

By using constant multiple rule,

By using power rule and chain rule,

Now the rate of change of the volume is also equal to the difference of what is being added and what is oozing out

Thus,

By equating both equations,

Since,

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