Question

(1 point) Find the length L and width W (with W≤L) of the rectangle with perimeter 100 that has maximum area, and then find the maximum area.

Answer 1

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:

$100 = 2L+2W\\L = 50-W\\A=LW=W*(50-W)\\A=50W - W^2$

The value of W for which the derivate of the area function is zero is the width that yields the maximum area:

$A=50W - W^2\\\frac{dA}{dW}=0=50 - 2W\\ W=25$

With the value of the width, the length (L) and the area (A) can be also be found:

$L=50-25 = 25\\A=W*L=25*25\\A=625$

Since the values satisfy the condition W≤L, the answer is:

Width = 25

Length = 25

Area = 625