Solve the problem.
-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?

Question

Quizplus · Accepted Answer

Let the length and width of the room be L and W respectively. 
According to the problem, perimeter = 2(L + W) = 180
L + W = 90
Area of the room, A = L*W
We need to maximize A. To do this, we use the fact that for a fixed perimeter, the maximum area of a rectangle is obtained when it is a square. 
So, L = W = 45
Therefore, the largest room that can be built has dimensions 45 ft x 45 ft and its area is 2025 sq. ft. Let the length of the room be x and the width be y. The perimeter is 2x + 2y = 180 or x + y = 90. We want to maximize the area, which is xy. We can use the equation x + y = 90 to solve for y in terms of x: y = 90 - x. Substitute this into the equation for the area: A = xy = x(90 - x) = -x^2 + 90x. This quadratic function has a maximum value when x = -b/2a = -90/-2 = 45. Therefore, the dimensions of the largest room that can be built are 45 ft by 45 ft, with an area of 45 x 45 = 2025 ft^2.

Solve the Problem. -A Carpenter Is Building a Rectangular Room with a Fixed