SOLUTION: a gardner is fencing off a rectangular area area with a fixed perimeter of 68 feet. What is the maximum area? A) 1156 ft2 B) 17 ft2 C) 289 ft2 D) 4.25 ft2 E) None

Algebra ->  Points-lines-and-rays -> SOLUTION: a gardner is fencing off a rectangular area area with a fixed perimeter of 68 feet. What is the maximum area? A) 1156 ft2 B) 17 ft2 C) 289 ft2 D) 4.25 ft2 E) None      Log On


   

Question 721573: a gardner is fencing off a rectangular area area with a fixed perimeter of 68 feet. What is the maximum area?
A) 1156 ft2
B) 17 ft2
C) 289 ft2
D) 4.25 ft2
E) None
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Easy to answer with only a little bit of thinking and without writing anything (except for squaring a number in your head).
Cut the 68 into four equal parts and then square the result.

A more analytical way to solve:

The perimeter is already set at 68 feet. The question asks, what is the largest area for this rectangular region, based on that given perimeter. Using x and y for length and width, we have these:

A=xy and 2x%2B2y=68. A is for AREA. Using the perimeter equation,
2y=-2x%2B68
y=-x%2B34
Substituting this into A equation,
A=x%28-x%2B34%29
highlight%28A=-x%5E2%2B34x%29, which shows that A has a maximum, which would be at the vertex of the graph of A.

Find the vertex! Put A(x) into standard form, and read the vertex point from standard form equation function A(x).
A%28x%29=%28-1%29%28x%5E2-34x%2B%2834%2F2%29%5E2%29-%28-1%29%2834%2F2%29%5E2
highlight%28A%28x%29=%28-1%29%28x-17%29%5E2%2B17%5E2%29

Vertex is at (17,289). This will obviously be a SQUARE SHAPE. 17+17+17+17=68.
Note that a rectangle has its maximum area when it is a square. This is why one could solve this problem almost entirely in ones head.