Find the length and width of a rectangle that has perimeter meters and a maximum area.
A)8 m; 24 m.
B)1m; 31 m.
C)16 m; 16 m.
D)17 m; 15 m.
E)20 m; 13 m.

Question

Find the length and width of a rectangle that has perimeter  meters and a maximum area.
A)8 m; 24 m. 
B)1m; 31 m. 
C)16 m; 16 m. 
D)17 m; 15 m. 
E)20 m; 13 m.

Quizplus · Accepted Answer

Let the length and width of the rectangle be L and W respectively. The perimeter is given by:
2L + 2W = 11 meters
L + W = 5.5 meters

The area of the rectangle is given by:
A = LW

We want to maximize A, given that L + W = 5.5. We can solve for one of the variables and substitute into the equation for A to get a function of just one variable. For example, we can solve for L:
L = 5.5 - W

Substituting into A, we get:
A = (5.5 - W)W = 5.5W - W^2

This is a quadratic function with a maximum at the vertex. The vertex occurs at W = -b/2a, where a = -1 and b = 5.5. Thus:

W = -5.5/-2 = 2.75

Substituting this value back into L + W = 5.5, we get:

L = 5.5 - 2.75 = 2.75

Thus, the rectangle has dimensions L = W = 2.75 meters. This corresponds to answer choice C.

Find the Length and Width of a Rectangle That Has