Answer:
<u>15 feet x 15 feet</u>
Step-by-step explanation:
I'm worried about Fido, so I'll use three different approaches to maximizing his play space: Graphing, Chart, and First Derivative. This is perhaps more than Fido wants, but it hopefully offers ideas on how these types of problems can be answered.
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Fido is limited to 60 feet of fence, which will form the perimeter. The area of a rectangle is (length)*(width). Let's set length to L and width to W. I'll use A for area.
Therefore A = L*W
The rectangle's perimeter will be 2L + 2W, and this is equal to 60 feet:
2L + 2W = 60
Let's rearrange this equation to isolate either W or L. I chose L:
L + W = 30
L = 30 - W
Now let's use this definition of L in the Area equation:
A = L*W
A = (30-W)*W
A = 30W - W^2
<u>Graphing</u>
We can graph this function (A = 30W - W^2) to find the maximum area. See the attached image. We find a maximum area of 225 ft^2 when the width is 15 feet. 15 feet would mean that Fido's roaming pen will be in the shape of a square; 15 feet for each side equals 60 feet total fence.
<u>Chart</u>
One can also find a maximum by calculating a variety of conditions set by L = 30 - W. I randomly picked lengths from 2 to 24 feet and calculated the resulting width and then the resultant area for each condition. See attached image. Although I did not pick 15 feet, we can see that the area increases until the length transitions from 14 to 16 feet. It reverses direction at that point. A closer inspection around those values will show that 15 feet is, in fact, the optimum value to produce a pen with the most area.
<u>First Derivative</u>
One could also take the first derivative of the equation A = 30W - W^2 and set it equal to zero. The fist derivative tells us the slope of the line at any point (for a value of W in this example). The slope is zero only when the curve hits a maximum.
A = 30W - W^2
A' = 30 -2W
0 = 30 - 2W
2W = 30
W = 15 feet
The slope is zero when the width is 15 feet. 15 feet is the optimum for creating a pen with the most area possible with 60 feet of fence.
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All three approaches yield the same result. Graphing is fun when you have access to DESMOS. Charts are fun if you like Excel. Derivatives can also be fun, mostly because I always ponder "Who had the audacity to think this might work?" (Didn't they have something more important to do? Raid a village, etc.?).
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Arf
