A max/min problem - Math Central

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Question from bailey, a student:

Hi there,

A right angled triangle OPQ is drawn as shown where O is at (0,0).
P is a point on the parabola y = ax – x^2
and Q is on the x-axis.

Show that the maximum possible area for the triangle OPQ is (2a^3)/(27)

It has been bugging me from the test.
Thanks :)

Hi Bailey

I expect the diagram looks something like this. I gave $Q$ the coordinates $(x,0).$

parabola and triangle

What is the y-coordinate of $P?$ That's the height of the triangle. What is the area of this triangle? Use the calculus you know to find the value of $x$ that maximizes the area. What is the area at for value of $x?$

Penny

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