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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

Gravel is being dumped from a conveyor belt at a rate of 30ft 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 10ft high?

Chapter 3.9, Problem 29E, Gravel is being dumped from a conveyor belt at a rate of 30ft 3 min, and its coarseness is such that

To determine

To find: The rate of change of the height of the cone shape pile whose base diameter and

height are always equal.

Explanation

Given:

Consider the gravel is being dumped from a conveyor belt at a rate as dVdt=30ft3/min.

The diameter of the base and height of the pile are always same.

Formula used:

(1) Chain rule: dydx=dydududx

(2) Volume of the cone of base radius r and height h:V=13πr2h.

Calculation:

Let V be the volume of the cone shape pile and h be the height.

Since height and diameter of the cone shape pile are always same. Thus, radius of the cone shape pile is h2 .as shown in the figure-1 given below.

Since V and h changes with the time t, V and h are the function of the time t.

Obtain dhdt when h=10ft.

Substitute r=h2 in V,

V=13π(h2)2h=112πh3

Differentiate V with respect to the time t.

ddt[V]=ddt[112πh3]dVdt=112π(3h2)dhdt

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