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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

A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a height r / 1 + π 2 above the surface of the liquid.

To determine

To show:

The center of the disk should be positioned at height r1+π2 above the surface of the liquid

Explanation

1) Concept:

Exposed wetted area = area of the disk  - area of arc OAB + area of triangle OAB  -  area of circle with center O and radius rcosθ

2) Calculation:

Exposed wetted area= area of the disk  - area of arc OAB + area of triangle OAB -  area of circle with center O and radius rcosθ

That is,

A=πr2-r2θ+0.5r2sin2θ-πrcosθ2

To maximize this area, let us find critical points of A. So differentiate it with respect to θ and equate it to zero.

dAdθ=0-r21+0.5r22cos2θ-πr2(-)sin2θ

Simplify to get

dAdθ=-r2+r2cos2θ+πr2sin2θ

Set dAdθ=0

-r2+r2cos2θ+πr2sin2θ=0

Factor out r2 from the first two terms.

-r2(1-cos2θ)+πr2sin2θ=0

Simplify.

-r22sin2θ+πr2sin2θ=0

Factor out r2

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