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Question 1174323: Point P(u, v) is in the first quadrant on the
graph of the line connecting the points (0,4)
and (4,0). A shaded triangular region is
shown in the diagram. For what point P will
the area of the shaded triangular region be a
maximum?
I tried to get a formula for finding the area for the triangle inside. I got 1/2(4-u)(4-v) but I'm not sure where to go from there.
There was also a similar question on the site with the diagram since I can't seem to post it on here
https://www.algebra.com/cgi-bin/display-illustration.mpl?tutor=lolxcry&illustration_name=Math
Answer by ikleyn(52748)   (Show Source):
You can put this solution on YOUR website! .
Notice that the line through these point is the line x + y = 4.
Regarding the legs u and v, we have therefore u + v = 4.
Now we can reformulate our problem.
We see that x + y = 4, or u + v = 4.
It means that 2(u+v) = 8.
So the question is: among all rectangles with the given perimeter of 8 units,
which one has the maximum area ?
It is one of classic minimax problems, and the answer is WELL KNOWN:
a square with the side length of 1/4 of the perimeter
gives a maximum area.
So, u = v = 2 gives the maximum area of the shaded triangle.
Solved.
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Do you understand everything in my solution/explanation ?
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