SOLUTION: 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

Algebra.Com
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.

----------------

Do you understand everything in my solution/explanation ?



RELATED QUESTIONS

Point P (u,v) is in the first quadrant on the graph of the line connecting the points... (answered by ikleyn)
Point P (u,v) is in the first quadrant on the graph of the line connecting the points... (answered by MathLover1,lolxcry)
For each list U below, come up with a non-trivial (not all zeros) list V that makes the... (answered by ikleyn)
Consider the following final tableau corresponding to a linear programming problem. x (answered by Edwin McCravy)
Hello tutor, This one question on my online homework assignment has been puzzling me for (answered by lwsshak3)
if tan 0 = -(3/4) and 0 line in the 2th quadrant, the value of csc 0... (answered by stanbon)
1)u×u>=0 and u×u=0 if and only if u=0 prove theorem? 2)Show that there are no vectors u... (answered by lynnlo)
Please help me ㅠㅠ help me 1)u×u>=0 and u×u=0 if and only if u=0 prove... (answered by lynnlo)
1)u×u>=0 and u×u=0 if and only if u=0 prove theorem? 2)Show that there are no vectors u... (answered by lynnlo)