1. ## Optimization and Modeling

The hypotenuse of a right triangle has one end at the origin and one end on the curve y = (x^2)*e^(-3x), with x > 0. One of the other two sides is on the x-axis, the other side is parallel to the y-axis. Find the maximum area of such a triangle. At what x-value does it occur?

All right, I know that I'll probably need A = (1/2)bh. I know I have to in some way find out where the curve intersects either the vertical side of the triangle or the hypotenuse to maximize the area? I think? But I don't know if i'm thinking the right way, and I don't know how to go about setting this up.

2. Hello, Jacobpm64!

It's easier than you think . . .

The hypotenuse of a right triangle has one end at the origin
and one end on the curve $y = x^2e^{-3x}$, with $x \geq 0.$
One of the other two sides is on the x-axis, the other side is parallel to the y-axis.
Find the maximum area of such a triangle. At what x-value does it occur?
Code:
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|                 *
|            *
|         *  |
|       *    |y
|    *       |
--*------------+---
|      x

The area of a triangle is: $A \,= \,\frac{1}{2}bh$

The area of this triangle is: . $A \:=\:\frac{1}{2}xy$ . . . where $y \:=\:x^2e^{-3x}$

So we have: . $A \:=\:\frac{1}{2}x\cdot x^2e^{-3x}\quad\Rightarrow\quad A \:=\:\frac{1}{2}x^3e^{-3x}$

Differentiate: . $A' \;= \;\frac{1}{2}x^2\!\cdot\!e^{-3x}(-3) + \frac{1}{2}\!\cdot\!3x^2\!\cdot\!e^{-3x} \;= \;\frac{3}{2}x^2e^{-3x}(-x + 1)$

Equate to zero: . $\frac{3}{2}x^2e^{-3x}(-x + 1)\;=\;0$

Set each factor equal to zero and solve.

. . $\frac{3}{2}x^2\,=\,0\quad\Rightarrow\quad x\,=\,0$ . . . This gives minimum area ... ha!

. . $e^{-3x} \,=\,0$ . . . but $\frac{1}{e^{3x}}$ can never equal zero.

. . $-x + 1 \:=\:0\quad\Rightarrow\quad x\,=\,1$ . . . There!

Maximum area occurs when $\boxed{x \,= \,1}$ and $y \,= \,1^2\!\cdot\!e^{-3} \,= \,\frac{1}{e^3}$

The maximum area is: . $A \:=\:\frac{1}{2}(1)\left(\frac{1}{e^3}\right) \:=\:\boxed{\frac{1}{2e^3}\text{ square units}}$

3. thanks!

Is that LaTeX that you used to format the text?

If so, what's the tag for it in these forums? I know the other forums I use are , and they don't seem to work here

4. Originally Posted by Jacobpm64
thanks!

Is that LaTeX that you used to format the text?

If so, what's the tag for it in these forums? I know the other forums I use are , and they don't seem to work here
This is what you use.