SOLUTION: Suppose you wanted to build area for your dog. You have 24 meters of fencing,each in 1 meter sections. What rectangular shape would produce the largest area for your dog?

Question 738857: Suppose you wanted to build area for your dog. You have 24 meters of fencing,each in 1 meter sections. What rectangular shape would produce the largest area for your dog?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

You are in luck. The given number of fence panels is divisble by 4 which means that it is possible to maximize the area without compromising so that the panels fit.

Let

represent the available perimeter, i.e. the total length of fencing that you have. Let

represent the length of your rectangle, and let

represent the width of the rectangle.

Since perimeter, length, and width are related thusly:

we can define length in terms of width and perimeter thus:

Since Area is length times width, using the above expression for

we can write a function that yields area as a function of width.

The astute student should recognize this function as a quadratic in standard form with coefficients

, and

. You should also note that, given the negative lead coefficient, the parabolic graph opens downwards meaning that the value of the function at the vertex is a maximum.

Using the formula for the

-coordinate of the vertex:

.

So the width, in terms of the available perimeter, that yields the greatest area is the perimeter divided by 4. That means that 2 times the width is the perimeter divided by 2. Subtracting the perimeter divided by 2 from the perimeter, that leaves the perimeter divided by 2 which represents 2 times the length. Hence, the length of the maximum area rectangle is also

.

Therefore, for a given perimeter, the maximum area rectangle that can be constructed is a square with sides that measure one fourth of the perimeter.

John

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