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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

Gravel is being dumped from a conveyor belt at a rate of 30 f t 3 / min , and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?

Chapter 2.8, Problem 29E, Gravel is being dumped from a conveyor belt at a rate of 30ft3/min, and its coarseness is such that

To determine

To find:

The rate at which height of pile is increasing.

Explanation

1. Concept:

We can express the radius in terms of height and then replace this value in the formula of volume of cone. Then differentiate with respect to time and find the required rate.

2. Formula:

i. Volume of cone,V=13πr2h

ii. Power rule of differentiation:

ddxxn=n*xn-1

iii. Chain rule of differentiation:

ddtfx=ddxfx*dxdt

3. Given:

dvdt=30ft3/minh=10 ft

4. Calculations:

It is given that diameter and height are always equal

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