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Question
A herpetologist is designing a rectangular experimental Iguana pen to fit in the 8' by 20' corner outside of a Komodo dragon pen. If only 42 feet of fencing is available for the Iguana pen, what dimensions should be used to obtain the largest area for the new pen? (Note: the fencing can extend, be the same length, or be shorter than the 8' wall and 20' wall, but all 42' of fencing must be used.)

Answer
Take x a the length on the 20' side and y as the length on the 8' side.

As x goes from 0' to 20', y goes from 25' down to 15'.
The perimeter is x + 2y - 8, which is always 42'.
The area goes from 0 ft to 300 ft.
The formula for area is A = x(25 - x/2) = 25x - x/2.
The derivative is 25 - x.
Setting this to 0 gives x = 25', which is outside the range.
This means the graph has a upwards slope across the region specified.
The maximum in this region is 300 ft, and that is at x = 20'.

As x goes from 20' to 27', y goes from 15' down to 8'.
The perimeter is 2x + 2y - 28, which is always 42'.
The area goes from 300 ft down to 216 ft.
The formula for area is A = x(35 - x) = 35x - x.
The derivative is 35 - 2x.
Setting this to 0 gives x = 17.5', which is outside the range.
This means the graph has a downward slope across the region specified.
The maximum in this region is 300 ft, and that is at x = 20'.

There is not any point looking at the next, since the maximum is between these two.
It is at x=20', y=15', and the area is 15' x 20' = 300 ft.

Just out of curiosity, the last region was done.  Here it is:

As x goes from 27' to 31', y goes from 8' down to 0'.
The perimeter is 2x + y - 20, which is always 42'.
The area goes from 216 ft down to 0 ft.
The formula for area is A = x(62 - x) = 62x - x.
The derivative is 62 - 2x.
Setting this to 0 gives x = 31', which is outside the range.
This means the graph has a downward slope across the region specified.
The area goes from 216 down to 0, and this is less than 300,  

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).

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