The campus of a college has plans to construct a rectangular parking lot on land bordered on one side by a highway. There are 760 ft of fencing available to fen

Question

(a) Express the length of the remaining side to be fenced in terms of x.
(b) What are the restrictions on x?
(c) Determine a function A that represents the area of the parking lot in terms of x.
(d) Determine the values of x that will give an area between 20 comma 000 and 40 comma 000 ftsquared.
(e) What dimensions will give a maximum area, and what will this area be?

Answer 1

 

(A)-(C)

 

X is the width and Y is the length. As these are dimensions , they must be positive and non-zero.

 

The Perimeter is 2X+Y = 760 ----> Y = 760-2X

 

The area is XY = X(760-2X) = A(x)

 

 

(D)

 

 20000 < XY < 40000

 

20000 < x(760-2x) < 40000

 

20000 < 760x - 2x^2 < 40000

 

-20000 > 2x^2 - 760x > -40000

 

0 > 2x^2 - 760x + 20000 > -20000

 

Per quadratic formula and examining the graph at www.desmos.com,

 

the values are x=28.4451 and x=351.555, which will produce an area in this range

 

(E) The max occurs at the average of the zeros of this quadratic function

 

The zeros of A(x) are x=0 and x= 380

 

The max area is then 190*(760-2*190) = 190* (760-380) = 190*380 = 72200