Use Newton\'s method to approximate the given number correct to eight decimal pl

Question

Use Newton's method to approximate the given number correct to eight decimal places.

6 49




and also: Use Newton's method to approximate the given number correct to eight decimal places. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 24 ft, find the value of x so that the greatest possible amount of light is admitted. (Give your answer correct to two decimal places.)

Explanation / Answer

(a)

x = sixth root of 49

x^6 = 49

That means we are looking for the zeros of the function
f(x) = x^6 - 49

f'(x) = 6x^5

Applying Newton's method to this problem leads to the iteration formula:
x(a+1) = x(a) - f(x a)/f'(x a)
= x (a) - (x(a)^6 - 49)/ ( 6x(a)^5)
= (5/6)x(a)+ (49/6 x(a)^5) -----(1)

Since we know that start 2^6 = 64
and (1.9)^6 = 47.04 so lets start with x=1.9
For this particular function you can start with any value. However so closer you you start to to final value so faster the iteration will converge.
From (1)

x(0) = 1.9
x(1) = (5/6)*1.9 + 49/(6*1.9^5) = 1.913153207
x(2) = (5/6)*(1.913153207) + 49/(6*(1.913153207)^5) = 1.912931247
Similarly
x(3) = 1.912931183
x(4) = 1.912931183
Stop iteration because no further change on first eight decimal place occur)

=>

cube root of 49 = 1.91293118

(b)

the rectangle has dimensions
width = 48/(4+pi)
height = 24/(4+pi)

Let x be the width, let y be the height.
Radius of semicircle = x/2. perimeter of semicircle = pi*x/2
area of semicircle = pi(x/2)^2/2 = pi*x^2/8

window perimeter = 24 = x + 2y + pi*x/2 = x(1+pi/2) + 2y
so 2y = 24 - x(1+pi/2)
y = 12 - x(1/2 + pi/4)
window area = xy + pi*x^2/8
A = x(12 - x(1/2 + pi/4)) + pi*x^2/8
A = 12x - x^2(1/2 + pi/4) + pi*x^2/8
A = 12x - x^2(1/2 + pi/8)
to maximize A, take the derivative with respect to x and set equal to 0.
A' = 12 - 2x(1/2 + pi/8) = 0
x(1 + pi/4) = 12
x = 12/(1 + pi/4)
= 48/(4 + pi)

and y = 12 - x(1/2 + pi/4) so
y = 12 - 48(1/2 + ?/4)/(4 + pi)
= 12 - 12(2+pi)/(4+pi)
= 12(1 -(2+pi)/(4+pi))
= 12(2/(4+pi))
= 24/(4+pi)

x = 48 / (4 + pi) = 48 / (4+3.14) = 48/7.14 = 6.72

y = 24 / (4 + pi) = 24 / (4+3.14) = 24/7.14 = 3.36

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