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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 θ.

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

Derivative rule:

Constant Multiple Rule:

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

ddθ[cf(θ)]=cddθ[f(θ)] (1)

Calculation:

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

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

sinθ=Opposite sideHypotenuse side=x10

Cross multiply the equation,

x=10sinθ

Differentiate the equation with respect to θ,

dxdθ=ddθ<

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