# Maximum area of a triangle

• May 18th 2009, 05:12 AM
chengbin
Maximum area of a triangle
Given the curve $\displaystyle y=e^x$, let A and B be the points on the curve at x=0 and x=1 respectively. Placing point P on curve y, somewhere between A and B, obtain the maximum value of the area of triangle PAB.

I just blanked out. I don't even know where to start.
• May 18th 2009, 05:50 AM
apcalculus
Quote:

Originally Posted by chengbin
Given the curve $\displaystyle y=e^x$, let A and B be the points on the curve at x=0 and x=1 respectively. Placing point P on curve y, somewhere between A and B, obtain the maximum value of the area of triangle PAB.

I just blanked out. I don't even know where to start.

So the points are (0, 1) and (1, e), after measuring the y-coordinates.

Let the coordinates of point P be (x, e^x). The area of the triangle is base times height over 2. Let us call AB the base, and the height will be the perpendicular extending from point P onto the segment AB. (Hope you're following so far.)

First we measure the distance AB using the distance formula:

$\displaystyle AB = \sqrt{(1-0)^2 + (e-1)^2}$

For the height, we need to first find the equation of the line perpendicular to AB that passes through P. The slope of this line is the negative reciprocal of the slope of AB, which is $\displaystyle \frac{1}{1-e}$

Remaining steps:
Find the equation of the line containing PH, where PH is the height and H is on AB.

Use this equation to find the coordinates of H. (Intersect the two lines, which algebraically means: set the equations equal.)

Finally, apply the distance formula to find PH and set up the formula for the area of the triangle. Remember to only consider critical number between 0 and 1, and justify that the area is an optimum of the type you want (i.e. max and not min)

Good luck!!
• May 18th 2009, 05:52 AM
chengbin
Why are you finding the distance AB? A and B are on the curve $\displaystyle y=e^x$. You need to use integrals to find the length of that curve.

Why do I have to find the height? Isn't the height 1?

I don't know what slope has to do with this question. Isn't this just another optimization problem?
• May 18th 2009, 05:56 AM
apcalculus
Quote:

Originally Posted by chengbin
Why are you finding the distance AB? A and B are on the curve $\displaystyle y=e^x$. You need to use integrals to find the length of that curve.

Why do I have to find the height? Isn't the height 1?

It's a triangle, so we need the length of the secant line, not the arc.

The height changes at every level. Remember you're find one of infinitely many possible triangles that yields the largest area. Think of P as 'sliding' along the exponential curve while connected with secant lines to both A and B.
• May 18th 2009, 06:27 AM
chengbin
I'm sorry, but do you mind showing me how to do this and get the answer?