Calculus
posted by Liz on .
The strength of a beam with rectangular corsssection 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

Strength, S = xy²
But we know that
x²+y²≤(3/2)²
therefore, substitute
S(x)=x((3/2)²x²)
For maximum strength,
dS(x)/dx = 9/43*x^2 =0
=>
x²=3/4
x=sqrt(3/4)
y^2=(3/2)^2x^2=9/43/4=6/4
y=sqrt(1.5)