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Chapter 3.3, Problem 37E
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### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall . If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when θ = π/3?

To determine

To find: The rate of change of x with respect to θ when θ=π3. That is, dxdθ|θ=π3.

Explanation

Given:

The length of the ladder is 10 ft, which rests against the vertical wall.

The angle between the top of the ladder and the wall is Î¸.

The distance from the bottom of the ladder to the wall is x.

Derivative rule:

Constant Multiple Rule:

If c is constant and f(Î¸). is differentiable function, then

ddÎ¸[cf(Î¸)]=cddÎ¸[f(Î¸)] (1)

Calculation:

The given situation is as shown in the below Figure 1.

In Figure 1, Î¸ is the angle between the top of the ladder and x is the distance from the bottom of the ladder to the wall.

sinÎ¸=OppositeÂ sideHypotenuseÂ side=x10

Cross multiply the equation,

x=10sinÎ¸

Differentiate the equation with respect to Î¸,

dxdÎ¸=ddÎ¸<

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Single Variable Calculus: Early Transcendentals