The strength of a beam with rectangular corss-section is directly proportional to the product of the width and the square of the depth (thickness from the top to bottom of the beam). Find the shape of the strongest beam that can be cut from a cylindrical log of diameter d= 3 ft

Answer 1

Strength, S = xy²
But we know that
x&sup2+y²≤(3/2)²
therefore, substitute
S(x)=x((3/2)²-x²)
For maximum strength,
dS(x)/dx = 9/4-3*x^2 =0
=>
x²=3/4
x=sqrt(3/4)
y^2=(3/2)^2-x^2=9/4-3/4=6/4
y=sqrt(1.5)