Question

Anyone can help me with this question? I would so much appreciate for helping me out!

Answer 1

I strongly recommend that you draw this situation.  The vertices of the right triangle are (2,0), (p,0) and (p, -p^2+8p-12).

We are to maximize the area of the triangle.  To do this, we find the equation for the area (which is based upon the usual b*h/2), differentiate this formula, set the derivative = to 0 and solve for p.  I obtained p = 14/3, or 4 2/3 units.

Keep in mind that the correct expression for the base here is p-2.  Can you see why?  The height is simply the function itself:  p(x) = -x^2+8x-12.
Thus, the area of the triangle, in terms of p, is

            (p-2)(-p^2+8p-12)
A(p) = ---------------------------
                           2

Let me know if you need further help with this problem.  Good luck!