Answer:
Width = 25
Length = 25
Area = 625
Step-by-step explanation:
The perimeter of a rectangle is given by the sum of its four sides (2L+2W) while the area is given by the product of the its length by its width (LW). It is possible to write the area as a function of width as follows:
[tex]100 = 2L+2W\\L = 50-W\\A=LW=W*(50-W)\\A=50W - W^2[/tex]
The value of W for which the derivate of the area function is zero is the width that yields the maximum area:
[tex]A=50W - W^2\\\frac{dA}{dW}=0=50 - 2W\\ W=25[/tex]
With the value of the width, the length (L) and the area (A) can be also be found:
[tex]L=50-25 = 25\\A=W*L=25*25\\A=625[/tex]
Since the values satisfy the condition W≤L, the answer is:
Width = 25
Length = 25
Area = 625
