SOLUTION: The dimensions of a rectangular piece of paper ABCD are AB= 10 and BC = 9. It is folded so that corner D is matched with point F on edge BC. Given that length DE= 6, find EF,EC,FC,

Question 1196071: The dimensions of a rectangular piece of paper ABCD are AB= 10 and BC = 9. It is folded so that corner D is matched with point F on edge BC. Given that length DE= 6, find EF,EC,FC, and the area EFC.
The lengths EF, EC, and FC are all functions of the length DE. The area of triangle EFC is also a function of DE. Using X to stand for DE, write formulas for these four functions,
Find the value of X that maximizes the area of triangle EFC.
Diagram link:
https://math.stackexchange.com/questions/2421004/areas-in-a-folded-sheet-of-paper

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The referenced URL tells you how you can answer the questions; did you try that?

EF is the same length as DE, so EF=x.

DC is 10, and DE+EC=DC, so EC=10-x.

Then use the Pythagorean Theorem to find an expression for FC:

FC =

And the area of EFC is one-half base times height:

To find the value of x that maximizes the area of triangle EFC, by far the easiest thing to do is graph the area function on a graphing calculator and use the calculator's features to find the answer.

Doing that shows the maximum area to be when x=20/3.

Or you could use calculus, if you know the subject....

(The factor of 1/2 in the area formula is not important for finding the maximum area, so I will ignore it in my work.)

Use the product rule and chain rule to find the derivative.

Find where the derivative is equal to zero.

The calculus confirms the answer found using a graphing calculator.

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