Question

Find the value of p that will provide the greatest área for the triangle.

Answer 1

Draw a picture of this situation.  Recognize that the base of the triangle is p-2 and that the height is -x^2+8p-12.  Thus, the area of this triangle is
        (b)(h)        (p-2)(-x^2+8p-12)
A = ---------- = -----------------------------
            2                      2

Differentiate this function.  Set the result = to 0.  Solve for the critical values 2 and 14/3.  Only p=14/3 makes sense, since the area of the triangle would be zero if you were to choose p=2.

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