Optimization Application

Question

A rectangular beam is cut from a cylindrical log of radius r = 48 cm. The strength of a beam of width w and height h is proportional to wh²

Answer 1

You want to optimize the strength given any constraints

S = wh²

The constraint is that the rectangle has to fit in the circle. Centering has to be right, as that maximizes both the width and the height. It must be true that a right triangle is formed with sides w/2, h/2 and hypotenuse of r. Therefore, (w/2)² + (h/2)² = 48² or w² + h² = 4*48² = 9216

It would be clever to substitute for h² = 9216 - w² into the strength equation.

Take the derivative with respect to w after distributing and set equal to 0. Solve for w using quadratic formula.

Solve for h, solve for S by plugging into the 2 equations.