Find the length and width of a rectangle that has perimeter 56 meters and a maximum area.
A) 7, 21
B) 1, 27
C) 14, 14
D) 15, 13
E) 18,10

Question

​Find the length and width of a rectangle that has perimeter 56 meters and a maximum area. ​​
A) ​7, 21
B) ​1, 27
C) ​14, 14
D) ​15, 13
E) ​18,10

Quizplus · Accepted Answer

Let the length and width of the rectangle be l and w respectively. 
From the problem statement, we have:
Perimeter = 2l + 2w = 56
Solving for l, we get: l = 28 - w
The area of the rectangle is given by: A = lw
Substituting the value of l from the first equation, we get:
A = (28 - w)w = 28w - w^2
This is a quadratic equation with a negative coefficient of w^2, meaning that the graph opens downwards. Therefore, the maximum value of the area occurs at the vertex of the quadratic, which is at w = -b/2a = -28/-2 = 14.
Substituting w = 14 into the first equation, we get: l = 28 - 14 = 14
Therefore, the length and width of the rectangle that maximize its area are 14 meters each. The correct answer is C.

​Find the Length and Width of a Rectangle That Has

Find the Length and Width of a Rectangle That Has