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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

The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle θ is increasing at a rate of 0.05 radian/s. How fast is the area of the triangle changing when x = 40 in, y = 50 in, and θ = π/6?

To determine

To find: The area of a triangle when x=40,y=50,θ=π6 .

Explanation

Given:

Let x and y be the side lengths of the triangle and the angle between the x and y is θ where dxdt=3in/s , dydt=2in/s and dθdt=0.05 .

The above information mentioned in the following Figure 1.

Calculation:

The area of a triangle is, A=12xysinθ .

By implicit differentiation,

dAdt=ddt[12xysinθ]=12[ysinθddt(x)+xsinθddt(y)+xyddt(sinθ)]=12[ysinθ(dxdt)+xsinθ(dydt)+xycosθ(dθdt)]

Substitute dxdt=3in/s , dydt=2in/s and dθdt=0

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Chapter 14 Solutions

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