SOLUTION: a point P is randomly chosen in the line AB of length 2a.what is the probability that the area of the rectangle having sides AP and PB will exceed a^2/2?

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Question 1030260: a point P is randomly chosen in the line AB of length 2a.what is the probability that the area of the rectangle having sides AP and PB will exceed a^2/2?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let x be such that 0+%3C=+x+%3C=+2a. Then the adjacent sides of the rectangle would have sides x and 2a - x. (Why?)
The area of the rectangle would be A%28x%29+=++x%282a+-+x%29.
The area under the curve is given by square units.
==> The cdf for the area is given by .
Now solve for the bounds on x that will give a rectangular area of a%5E2%2F2:
A%28x%29+=++x%282a+-+x%29+=+a%5E2%2F2
<==> 0+=+x%5E2+-+2ax+%2B+a%5E2%2F2 after simplifying.
==> x+=+%28%282-sqrt%282%29%29%2F2%29a, x+=+%28%282%2Bsqrt%282%29%29%2F2%29a.
==>
=.
This is approximately equal to highlight%280.88388%29 to five significant figures.