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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow.

Chapter 10.2, Problem 74E, A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the

To determine

To find: The grazing area available for the cow tied to a silo with radius r .

Explanation

Given:

When a cow is tied to a silo with radius r . The rope is long enough to reach the opposite side of the silo.

The value of θ varies from 0 to π .

Calculation:

A portion of involute is traced when cow walks with the rope taut as shown in below figure 1.

When θ=π the arriving point of cow is (r,πr) .

The cow walks in semicircle of radius πr when fully extended with the rope. The cow traces another portion of the involute, it is the reflection about the x axis of the initial involute path.

The value of θ ranges from π to 0 .

From Figure 1, the total grazing area is 2(A1+A3) .

Here, A3 is the area of a circle.

The area A3 is one quarter of the area of a circle of radius πr .

Calculate the area of the quarter circle A3 .

A3=14π(πr)2=14π(π2r2)=14π3r2

The total area of the involute (A1+A2) is calculated by applying integral over the limits of the involute.

The right most point of the involute is (πr2,r) and the left most point of the involute is (r,πr) .

Calculate the total area of the involute (A1+A2) .

A1+A2=θ=ππ2ydxθ=0π2ydx=θ=ππ2ydx+π2θ=0ydx=θ=π0ydx . (1)

Therefore,

ydx=r(sinθθcosθθ2cos2θ)dθ=r2(θsinθcosθθ2cos2θ)dθ

Integrate the above equation on both sides

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