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,

Algebra ->  Functions -> 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,      Log On


   

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) About Me  (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 = sqrt%28%28x%5E2%29-%2810-x%29%5E2%29=sqrt%28x%5E2-%28x%5E2-20x%2B100%29%29=sqrt%2820x-100%29

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

%28%2810-x%29%28sqrt%2820x-100%29%29%29%2F2

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.)

y=%2810-x%29%28%2820x-100%29%5E%281%2F2%29%29

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




Find where the derivative is equal to zero.

%28-%2820x-100%29%5E%281%2F2%29%29%2B%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29=0

%28%2820x-100%29%5E%281%2F2%29%29=%2810%2810-x%29%2F%2820x-100%29%5E%281%2F2%29%29

%2820x-100%29=10%2810-x%29
20x-100=100-10x
30x=200
x=200%2F30=20%2F3

The calculus confirms the answer found using a graphing calculator.