Question 68: A field bordering a straight stream is to be inclosed. The side bordering the stream is not to be fenced. If 1000yards of fencing material is to be used, what are the dimensions of the largest rectangle field that can be fenced What is the maximum area?
Answer by mhapich(1)   (Show Source):
You can put this solution on YOUR website! You want to start with a picture, in which one side of the field is the stream, and the other three sides of the field are labeled. In my picture, I have the stream on the left, and the top and bottom labeled with x and the right side labeled with y. Now, you write an equation with the given perimeter:
Next, you write an equation for the area, which you are trying to maximize:
Now, solve the perimeter formula for y (the easier variable for which to solve):
Subsitute that in for y in the area equation:
Simplify:
Now, take the derivative of the area function. Remember, taking the derivative and setting it equal to zero, then testing intervals surrounding this value for signs, will tell where the area function increased then decreased, thus finding its maximum.
Setting dA/dx equal to zero, we get:
Now, test values on either side. Plugging 1 into the derivative:
which is positive, so A is increasing on all values of x up to 250.
And, plugging 300 into the derivative:
which is negative, so A is decreasing on all values of x after 250.
Thus, the desired dimensions are 250 feet by 500 feet.
(Plug 250 in for x back into the above equation:
 ).
Also, remember to answer all questions -- "What is the maximum area?"
The area is
which will give you 250 * 500, or 125000 square yards.
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