A carpenter is building a rectangular room with a fixed perimeter of 180 ft. What are the dimensions of the largest room that can be built? What is its area?

45ft x 45ft

Step-by-step explanation:

we know that

Perimeter = 2(Length) + 2(Width)  

180 = 2L + 2W

90 = L + W

90 - L = W

Area = Length *Width

= L(90 - L) →  = 180L - 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.

= 90L - L² → 0 = 90 - 2L → 2L = 90 → L = 45

Algebra: find the vertex.

= - L² + 90L → L = -b/2a → L = -90/2(-1) → L = 45

insert L it into the Perimeter equation to solve for W:

90 - L = W → 90 - 45 = W → W = 45

45ft x 45ft