Question

A carpenter is building a rectangular room with a fixed perimeter of 300 ft. what dimensions would yield the maximum​ area?

Answer 1

Perimeter = 2(Length) + 2(Width)

300 = 2L + 2W → 150 = L + W → 150 - L = W

Area = Length · Width

$A_{max}$ = L(150 - L) → $A_{max}$ = 150L - L²

There are 2 different methods for solving (depending on which math course you are taking).

Calculus: find the derivative, set it equal to zero, and solve for L.

$A_{max}$ = 150L - L² → 0 = 150 - 2L → 2L = 150 → L = 75

Algebra: find the vertex.

$A_{max}$ = - L² + 150L → L = -b/2a → L = -150/2(-1) → L = 75

Once you solve for L, plug it into the Perimeter equation to solve for W:

150 - L = W → 150 - 75 = W → W = 75

Answer: 75ft x 75ft