rectangular beam optimization

Question

A rectangular beam is cut from a cylindrical log of radius r = 33 cm. The strength of a beam of width w and height h is proportional to wh².

find the width and height of the beam of maximum strength. round 3 decimal places

w = ?

h = ?

Answer 1

This is to maximize the strength of a rectangle enclosed in a circle of radius 33 cm.

Let w be the width and h be the height of the rectangle.

Draw a rectangle with the edges touching the circle. Draw the radius from the center of the circle to one of the corners.

h/2 = 33 sin(θ)

w/2 = 33 cos(θ)

Let S be the strength

S=wh²=66cos(θ)(66sin(θ))² = 66³cos(θ)sin²(θ))

We need to maximize θ

S'(θ)=66³(2cos²(θ)sin(θ)-sin³(θ))=66³sin(θ)(2cos²(θ)-sin²(θ))=0

2cos²(θ)=sin²(θ) -->tan²(θ)=2

sin(θ) = 0 -->θ = 0,π/2

tan(θ)=√2 --> θ=tan^-1(√2)

w=66cos(tan^-1(√2))=38.105 cm

h=66sin(tan^-1(√2))=53.889 cm

Using θ = 0,π/2 makes w=66 and h=0, which isn't useful.