Question

Consider the proft function: P(r)--212+420 (a) What is the average rate of change as the number of units produced increases from 9500 5 to 10? (b) Find the coordinates of the vertex and interpret what each coordinate represents.

Answer 1

The profit function is .

a) The average rate of change is

$\frac{\Delta P}{\Delta x}=\frac{P\left ( x_2 \right )-P\left ( x_1 \right )}{x_2-x_1}\\ \frac{\Delta P}{\Delta x}=\frac{P\left ( 10 \right )-P\left ( 5\right )}{10-5}\\ \frac{\Delta P}{\Delta x}=\frac{-2.1\times 10^2+420\times 10-9500-\left ( -2.1\times 5^2+420\times 5-9500 \right )}{5}\\ {\color{Blue} \frac{\Delta P}{\Delta x}=388.5}$

b) In vertex form the equation $P\left ( x \right )=-2.1x^2+420x-9500$ becomes,

$P\left ( x \right )=-2.1\left (x^2-200x \right )-9500\\ P\left ( x \right )=-2.1\left (x^2-200x +10000 \right )+21000-9500\\ P\left ( x \right )=-2.1\left (x-100\right )^2+11500$

The interprestation is the maximum profit is , when 100 units are sold.