Kyle P.
asked 01/01/20Applied Max and Min
A farmer has 1,200 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. Write the function that will produce the largest area if x is the short side of the rectangle.
Options:
- f(x) = 1200x - x2
- f(x) = -2x2 + 1200x
- f(x) = x2 - 1200
- f(x) = 2x2 - 1200
1 Expert Answer

Daniel M. answered 01/02/20
Passionate Chemistry and Calculus Tutor w/ 6+ Years Experience
To solve this, we'll need to define the area and the perimeter in terms of variables x and y.
Drawing a picture may also help. I will do my best to illustrate the rectangular area with the limited text options available here:
river
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x | | x
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y
The top horizontal line of the rectangle representing the river will not need fencing. Only sides x and y will need the fencing.
Now, defining the perimeter in terms of x and y of our depicted fenced in area would be:
Perimeter = 2x + y (1)
Next, defining the area in terms of x and y would be:
Area = xy (2)
Since we have 1200 ft of fencing, and the fencing is represented as the perimeter:
Perimeter = 1200
Now, we can substitute this number into equation (1):
1200 = 2x + y (3)
Since the question wants the area expressed as a function of x, we'll need to rearrange the above equation (3) and substitute it into equation (2):
1200 = 2x + y ⇒ y = 1200 – 2x
Let f(x) = the area of the rectangular space:
f(x) = Area ⇒ f(x) = xy
Substituting y from the rearranged equation (3) into f(x) will yield:
f(x) = x(1200 – 2x)
Distributing then yields: f(x) = 1200x – 2x2 which is equivalent to f(x) = –2x2 + 1200x
So answer is option 2
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Peter K.
Kyle, I am excited by your homework and would be pleased to answer one more for you this evening if there is a particular one that you tried and are stuck on but I am not sure that this ask an expert feature is designed to generate solution sets to assignments.01/01/20