   Chapter 3.9, Problem 25E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find t he rate at which water is being pumped in to the tank.

To determine

To find: The rate at which water is being pumped into the tank.

Explanation

Given:

The rate at which the water is leaking out from the inverted conical tank is 10,000 cm3/min.

The water is being pumped at a constant rate into the same tank.

The height of the tank is 6 m.

The diameter of the top is 4 m.

The net rate by which the water level increases of the tank is 20 cm/min.

Formula used:

(1) Chain rule: dydx=dydududx

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

Calculation:

Let V be the volume of the inverted cone and r be the radius of the top and h be the height of the cone, which is represented in the below Figure 1.

Let C be the rate by which the water is being plumbed into the tank.

Then, the net rate of change of volume of the water inside the cone is dVdt=C10,000cm3/min.

The rate of change of height of the water inside the tank is dhdt=20cm/min.

Obtain C when height of the water inside the tank is 2 m.

Eliminate r from V that is find r in terms of h

By the properties of similar triangles

r2=h6r=13h

Substitutes r=13h in V,

V=13π(13h)2×h=127πh3cm3

Differentiate V with respect to the time t,

ddt[V]=ddt[127πh3]dVdt=127π(3h2)dhdt&

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