I've been having trouble with this problem for the past few days for my homework. No matter what I try, I still can't solve it correctly. Please help!

1. An artist has come to you to help them determine where to place the "other" vertex of a right triangle. One acute angle will be on the x-axis (within the interval wherey0), the right angle will be on the x-axis, the other acute angle will be on the curvey=-0.1e^x(x^2-9)wherey0. The artist wants the triangle with the largest area noting that the triangle may not entirely be under the curve. At what x-coordinate do they place the right angle? Please use the steps below to help answer this question. Show ALL work.

• Sketch at least 3 possible situations and calculate the area of the triangles for each situation.
• What quantity do you want to optimize?
• What equation relates the quantity you want to optimize to other "variables" (these are actually quantities)? This should be a standard formula you learned in high school or before. Define variables for every quantity you have identified as important so far.
• What is the interval in the domain ofywherey0? This is the only interval where the two vertices on the x-axis can lie.
• What is the constraint equation? There are actually 2 of them, one for one of the dimensions, one for the other. (Hint1: Use the curve equation for one of the dimensions. Why?) (Hint2: Notice that the variable representing the quantity of the base length you need is not justx.)
• Using the two constraint equations rewrite the equation from part c so that you have a function of one variable. What is the domain of this function? It may (or may not) be the same as the pertinent interval of the curve from part d.
• Now, calculus time. Take a derivative, set to zero, find critical number(s).
• Test around the critical number(s) to see where (if) you have a maximum.
• Answer the question asked in a sentence.