Average Rate of Change

Question

A theatre has determined that its profit function is given by P(x) = x^2+9x-11, where x is the admission ticket price, in dollars. What would be the average rate of change in profit if the ticket price were to increase from $2 to $5.

Answer 1

Hi Ann,

The "average rate of change" means the same thing as "slope".

When the ticket price is $2, the profit is

P(2) = 2^2 + 9(2) - 11

= 4 + 18 - 11

= 11 dollars

This corresponds to a the point (2, 11)

When the ticket price is $5, the profit is

P(5) = 5^2 + 9(5) - 11

= 25 + 45 - 11

= 59 dollars

This corresponds to a the point (5, 59)

So when the price increases by $3, the profit increases by $48.

The slope of the line through (2, 11) and (5, 59) gives us the average change in profit as the ticket price changes from 2 to 5 dollars.

average rate of change = (difference of the y-coordinates) divded by (difference of the x-coordinates)

= (59 - 11) / (5 - 2)

= 48 / 3

= 16

So, on AVERAGE, the profit increased by $16 for each dollar the ticket price increased.

Hope this helps,

Jim :)