BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 24P
To determine

The water level raising at the instant the cone is completely submerged.

Expert Solution

Explanation of Solution

Given information:

A cone of radius r and height h is lowered point first at a rate of 1cm/s in a tall cylinder of radius R cm which is partially filled with water.

Calculations:

Let a be the depth that the cone is submerged.

We are given that dadt=1

The volume of water that is displaced is given by the formula V=13πb2a ,

where b is obtained from the relation

  hr=abor,b=arh

Thus V=13π(arh)2a

Now differentiating the volume we get,

  dVdt=πr2a2h2.dadt

When the cone is completely submerged, a=h and we have dVdt=πr2 .

Let l denote the water level in the cylinder.

Then the volume of water in the cylinder is given by V=πR2l ,

and we have dVdt=πRdldt2 .

Putting in the value of dVdt , and solving for dldt , we get,

  dldt=r2R2 .

Thus we can say that the water level is rising at a rate of r2R2 cm/s at the instant the cone is completely submerged.

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