c(x)=5x + 24, p(x)=(-2x^2 + 27x)-(5x + 24)=-2x^2+22x - 24. b) determine the number of telephones sold that…

Explanation:

Step1: Recall profit - maximization condition

The profit function is $P(x)= - 2x^{2}+22x - 24$. For a quadratic function $y = ax^{2}+bx + c$ ($a\neq0$), the x - value of the vertex (which gives the maximum or minimum) is found using the formula $x=-\frac{b}{2a}$. Here, $a=-2$, $b = 22$, $c=-24$.

Step2: Calculate the number of telephones for maximum profit

Using the formula $x=-\frac{b}{2a}$, we substitute $a=-2$ and $b = 22$ into it. So $x=-\frac{22}{2\times(-2)}=\frac{22}{4} = 5.5$.

Step3: Calculate the maximum profit

Substitute $x = 5.5$ into the profit function $P(x)=-2x^{2}+22x - 24$. $P(5.5)=-2\times(5.5)^{2}+22\times5.5-24$ $=-2\times30.25 + 121-24$ $=-60.5+121 - 24$ $=36.5$.

Answer:

The number of telephones sold to maximize profit is $x = 5.5$ (in thousands), and the maximum profit is $36.5$ (in thousands).