   Chapter 4.8, Problem 40E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### 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. 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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