How to find the maximum area of a rectangle?

Question

Ramon wants to fence in a rectangular portion of his back yard against the back of his garage for a vegetable garden. He plans to use 68 feet of fence, and needs fence on only three sides. Find the maximum area he can enclose. (Hint: The lengths of the 3 fenced sides of the rectangle must add up to 68.)

Answer 1

Let y = length of side that is parallel to the side without fencing

     x = length of each of the other two sides

 

Then, y+2x = 68

 

            So, y = 68-2x

 

Area = xy = x(68-2x)

 

 The graph of the area function is a parabola opening downward.  The x-intercepts are (0,0) and (34,0).  By the symmetry of the parabola, the maximum area occurs halfway between the x-intercepts.  So, the area is maximized when x = 17 and y = 68-2(17) = 34.

 

Maximum area = (17)(34) = 578 ft²

 

Answer 2

**NOTE**   THX to Andrew. M., a great problem-solver and proof-reader for the comments (I have edited my answer but WyzAnt Answers Forum doesn't allow him to change his comments.)

The area is the length times the width. The length could be large and the width small (see table below) [e.g., 66*1 (note 66+1+1=68 feet of fence)] or the length could be small and the width large [e.g., 2*33 (that means 2+33+33=68)]. Also, the garage has a limited size (but we can ignore that).

As the length decreases the width must increase (there is exactly 68 feet of fence). So, values like 66*1 (area=66) change to values like 50*9 (area=450) ... to the value 34*17 (area=578) … to values like 18*25 (area=450) to values like 2*33 (area=66). The area goes up then back down. The maximum area occurs at L=34 and W=17 (A=578).

[Note: you may also include fractions on either side of (34,17) to test that the maximum occurs at (34,17).]

68 = 2L + 1W
Length Width Area
   66        1        66
   64        2      128
   62        3      186
   60        4      240
   58        5      290
   56        6      336
   54        7      378
   52        8      416
   50        9      450
   48      10      480
   46      11      506
   44      12      528
   42      13      546
   40      14      560
   38      15      570
   36      16      576
   34      17      578
   32      18      576
   30      19      570
   28      20      560
   26      21      546
   24      22      528
   22      23      506
   20      24      480
   18      25      450
   16      26      416
   14      27      378
   12      28      336
   10      29      290
     8      30      240
     6      31      186
     4      32      128
     2      33        66