Question

Conor is going to the movie theater. A ticket to a movie costs t dollars, and there is a 15% amusement tax on each ticket.

1. Conor buys a ticket to see a movie. Write two expressions in terms of t showing his total cost.
2. choose a value for t greater than $12 and less than $13. Use one of the expressions you wrote in part 1 to find Conor's total cost, rounded to the nearest penny. Show your work.
3. suppose that Conor's total cost to see the movie is greater than $17.50 and less than $20. What could be a value of t to make that possible?

Answer 1

1. To find Conor's total cost for a ticket, we need to consider both the ticket price and the amusement tax. The amusement tax is 15% of the ticket price \( t \). Thus, we can express the total cost in two ways:

- Total cost without tax: \( C = t \)
- Total cost including the tax: \( C = t + 0.15t \)

We can simplify the second expression:

\[
C = t + 0.15t = 1.15t
\]

Therefore, the two expressions for Conor's total cost in terms of \( t \) are:
- \( C = t \)
- \( C = 1.15t \)

2. Let's choose a value for \( t \) between $12 and $13. For instance, we'll choose \( t = 12.50 \).

Now we can use the expression for the total cost including tax:

\[
C = 1.15t
\]

Substituting \( t = 12.50 \):

\[
C = 1.15 \times 12.50
\]

Now, let's calculate:

\[
C = 1.15 \times 12.50 = 14.375
\]

Rounding to the nearest penny:

\[
C \approx 14.38
\]

Thus, Conor's total cost is **$14.38**.

3. We are told that Conor's total cost \( C \) is greater than $17.50 and less than $20. So, we can set up the inequality using the total cost including tax:

\[
17.50 < 1.15t < 20
\]

To solve for \( t \), we can divide the entire inequality by 1.15:

\[
\frac{17.50}{1.15} < t < \frac{20}{1.15}
\]

Now, we calculate the bounds:

\[
\frac{17.50}{1.15} \approx 15.2174
\]
\[
\frac{20}{1.15} \approx 17.3913
\]

Therefore, the inequality simplifies to:

\[
15.2174 < t < 17.3913
\]

With the condition that \( t \) must be greater than $12 and less than $13, we can find values greater than 15.2174 and less than 17.3913.

A valid value for \( t \) that meets this criterion could be \( t = 16.00 \).

Thus, a possible value for \( t \) is **$16.00**.