91Ó°ÊÓ

The equation can be given as

$$\begin{gathered} R\left(x\right)=9.5x-0.04x^2 \\ C\left(x\right)=1.25x+250. \end{gathered}$$

in the revenue equation the coefficient of $x^2$term is negative. hence the parabola opens downwards. the maximum value is at the vertex

The vertex can be given as

$$\begin{gathered} x=-\frac{b}{2a} \\ x=\frac{9.5}{0.08} \\ x=118.75 \end{gathered}$$

The corresponding revenue is

$$\begin{gathered} R(118.75)=9.5(118.75)--0.04{(118.75)}^2 \\ R(118.75)=564.06 \end{gathered}$$

We get

$$\begin{gathered} P\left(x\right)=9.5x-0.04x^2-1.25x-250 \\ P\left(x\right)=-0.04x^2+9.25x-250 \end{gathered}$$

As the coefficient of $x^2$term is negative the parabola opens downward hence the maximum value is at the vertex

The vertex can be given as

$$\begin{gathered} x=-\frac{b}{2a} \\ x=\frac{8.25}{0.08} \\ x=103.125 \end{gathered}$$

The maximum profit is

$P(103.125)=175.39$

As the function of profit depends on the cost function and it does not depend on revenue function Hence the maximum values are different

a)To maximize the revenue 119 boxes should be sold and revenue is $564

b)The profit equation is $P\left(x\right)=-0.04x^2+8.25x-250$

c)to maximize the profit 103 boxes should be sold and the profit is $175

d)As the function of profit depends on the cost function and it does not depend on revenue function Hence the maximum values are different