\(TP(p) = -15p^2 + 195p - 450\)
\(a) \text{find }TP(4.5)\)
\(b) \text{at this intersection }TP(p) = 0\\ \text{This is a simple quadratic equation. Solve it for it's roots. }\\ \text{Use the quadratic formula if you have to}\)
\(c)~\text{0 profit means why bother}\)
\(d)~\text{We want to find the maximum value of TP.}\\ \text{As }TP(p) \text{ is a quadratic equation it's graph will be a parabola}\\ \text{In this case as the coefficient of the }p^2 \text{ term is negative it will be facing downwards}\\ \text{and thus the maximum value will occur at the vertex of the parabola}\\ \text{A quick formula for the }x \text{ coordinate of the vertext of }a p^2 + bp + c \\ \text{is }p = -\dfrac{b}{2a} \text{, in this case }p = -\dfrac{195}{2(-15)} = \dfrac{13}{2}\\ \text{Plug this value into }TP(p) \text{ and read off the maximum profit that can be achieved}\)
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+6252 