Question

3. A rock club's profit from booking local bands depends on the ticket price, Using past receipts, the owners find that the profit p can be modeled by the function p = -10t + 500t + 60, where t represents the ticket price in dollars, a) If the ticket price is $16, how much money does the club earn in profits? [2]b) What price yields the maximum profit? [2]c) What is the maximum profit? [2]

Answer 1

`p=-10t^2\text{ + 500t + 60}`

Part A

To find the ticket price when the price is $16

Let us substitute the value of t = 16

p = -10 x (16 x16) + 500 x 16 + 60

p = -2560 + 8000 + 60

p =$ 5500

Part B

To get the maximum profit, we will have to differentiate P with respect to t

`\begin{gathered} p=-10t^2\text{ + 500t + 60} \\ \frac{dp\text{ }}{dx}=\text{ -20t + 500} \end{gathered}`

The maximum profit will be obtained when the derivative is zero

-20t + 500 = 0

20t = 500

t = 500/20

t = 25

This means that the ticket price has to be $25 so as to obtain the maximum price

Part C

The maximum profit will be obtained by substituting t = 25 into the original equation

`\begin{gathered} p=-10t^2\text{ + 500t + 60} \\ p\text{ = -10 x 25 x 25 + 500 x 25 + 60} \\ p\text{ =}6310 \end{gathered}`