5. The Valley High School student council is planning a dance. The school has 135 students. The lowest possible ticket price is $\mathrm{\$}1.00$, and they estimate that for every $\mathrm{\$}0.50$ increase in ticket price, 15 fewer students will attend. What ticket price will maximize the student council's profit?
$\mathrm{\$}3.15$
$\mathrm{\$}2.65$
$\mathrm{\$}2.50$
$\mathrm{\$}2.75$

Step 1 ：The Valley High School student council is planning a dance. The school has 135 students. The lowest possible ticket price is $1.00,andtheyestimatethatforevery$0.50 increase in ticket price, 15 fewer students will attend. What ticket price will maximize the student council's profit?

Step 2 ：The profit is calculated by multiplying the ticket price by the number of students attending. The number of students attending is a function of the ticket price.

Step 3 ：We can create a function to calculate the profit for each possible ticket price and then find the maximum profit.

Step 4 ：Let's consider the following ticket prices: $1.00,$1.50, $2.00,$2.50, $3.00,$3.50, $4.00,$4.50, $5.00,$5.50.

Step 5 ：The corresponding profits for these ticket prices are: $135.00,$180.00, $210.00,$225.00, $225.00,$210.00, $180.00,$135.00, $75.00,$0.00.

Step 6 ：The maximum profit is obtained for the ticket price of $2.50.

Step 7 ：Final Answer: The ticket price that will maximize the student council's profit is $\boxed{{\displaystyle \mathrm{\$}2.50}}$.