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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

In the figure, the length of the chord AB is 4 cm and the length of the arc AB is 5 cm. Find the central angle θ, in radians, correct to four decimal places. Then give the answer to the nearest degree.

Chapter 4.8, Problem 40E, In the figure, the length of the chord AB is 4 cm and the length of the arc AB is 5 cm. Find the

To determine

To Find: The central angle in radian of the circle correct to the four decimal places and give the answer to the nearest degree.

Explanation

Given:

The length of the chord line is 4 cm.

The length of the arc is 5 cm.

Result used: (Newton’s method).

(a)θn+1=θnf(θn)f(θn) Where f(θ) is function of θ.

(b)s=rθ, Where s= arc length,r= radius of the circle and θ= central angle.

(c)180=πc

(d)Law of cosine cosθ=b2+c2a22bc

Calculation:

From the result (b) set s=5 and calculate radius r=5θ

From result (d) set r=4,a=r and b=r.

42=r2+r22×r×r×cosθ42=2r22r2cosθ42=2r2(1cosθ)

Set r=5θ to calculate the above expression.

42=2(5θ)2(1cosθ)42=2(25θ2)(1cosθ)16θ2=50(1cosθ)16θ2+50cosθ50=0

Let f(θ)=16θ2+50cosθ50.

Calculate the derivative of the above function f(θ).

Let the derivative of f(θ) is f(θ)=df(θ)dθ.

f(θ)=d(16θ2+50cosθ50)dθ=16d(θ2)dθ+50d(cosθ)dθd(50)dθ=32θ50sinθ0=32θ50sinθ

Graph:

From the graph the initial point θ1=2.2.

The value of f(θ1) at the point θ1=2.2 is,

f(2.2)=32×2.250×sin(2.2)=70.450×0.808496=70.440.424820=29.975179

Calculate the value of f(θ1) at the point θ1=2

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