SOLUTION: A farmer wants to enclose a rectangular garden, with the plot to be divided into 3 equal sections. If he uses 400 feet of fencing, what is the maximum area the garden can be?

Question 1042045: A farmer wants to enclose a rectangular garden, with the plot to be divided into 3 equal sections. If he uses 400 feet of fencing, what is the maximum area the garden can be?

________ft2
* I honestly cannot figure out how to get this problem as there are no width or length measurements to work with.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Of course there is a length measurement to work with: the 400-foot perimeter. And since both the area and the perimeter can be expressed in terms of the length and width, you have the opportunity to create a non-linear 2X2 system, and then by substitution create a function for area in terms of width that you can maximize.

Draw a rectangle to represent the overall garden. Since the labels on the sides are arbitrary, label the shorter side

and the longer side

. Now draw in the two pieces of fence on the inside of the rectangle that divide it into three equal areas. It doesn't really matter to which outside edge they are parallel, but just so we are on the same page, make them parallel to the sides labeled

. Now you have 6 pieces of fence, 4 labeled

and 2 labeled

. Since he is going to use all 400 feet of fence we can write:

Which can be rearranged to:

Next, we know that the overall area of the garden is given by

, but we can substitute from the relation above to get:

Now you can either use the idea that this is a quadratic function with a negative lead coefficient so its graph is a parabola that opens downward and the value of the function at the vertex is a maximum, or you can find the zero of the first derivative and then evaluate the function at that value.

John

My calculator said it, I believe it, that settles it

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