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.
a. Conor buys a ticket to see a movie. Write two expressions in terms of t
showing his total cost.....................

Answer 1

Choose a value for t greater than $12 and less than $13. Use one of the
expressions you wrote in part a to find Conor’s total cost, rounded to the
nearest penny. Show your work./////////////////////////////////////

Answer 2

c. 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 3

To express Conor's total cost for a movie ticket considering the 15% amusement tax, we can break it down as follows:

1. **Cost of the ticket before tax**: This is simply \( t \) dollars.

2. **Amount of tax**: The amusement tax is 15% of the ticket price, which can be expressed as:
\[
\text{Tax} = 0.15 \times t
\]

3. **Total cost**: The total cost, including the ticket price and the tax, can be expressed as:
\[
\text{Total Cost} = t + 0.15t
\]

Alternatively, we can factor out \( t \):
\[
\text{Total Cost} = t(1 + 0.15) = t(1.15)
\]

Thus, the two expressions for Conor's total cost in terms of \( t \) are:
1. \( t + 0.15t \)
2. \( 1.15t \)

Answer 4

Let's choose a value for \( t \) within the specified range. We can use \( t = 12.50 \) dollars.

Now, we will use the expression for total cost that includes the amusement tax. We can use either of the expressions from part a, but let's use the second one for simplicity:

\[
\text{Total Cost} = 1.15t
\]

Substituting \( t = 12.50 \):

\[
\text{Total Cost} = 1.15 \times 12.50
\]

Now, calculate this value:

\[
\text{Total Cost} = 1.15 \times 12.50 = 14.375
\]

Now we round to the nearest penny:

\[
\text{Total Cost} = 14.38
\]

Thus, Conor’s total cost for the movie ticket, rounded to the nearest penny, is **$14.38**.

Answer 5

To find a value of \( t \) such that Conor's total cost lies between $17.50 and $20, we can use the expression for total cost that we derived earlier:

\[
\text{Total Cost} = 1.15t
\]

We want to determine the values of \( t \) that satisfy the following inequality:

\[
17.50 < 1.15t < 20
\]

Let's solve the inequalities step by step.

1. **For the lower bound**:

\[
17.50 < 1.15t
\]

Dividing both sides by 1.15:

\[
t > \frac{17.50}{1.15}
\]

Calculating the right side:

\[
t > 15.2174 \quad (\text{approximately})
\]

2. **For the upper bound**:

\[
1.15t < 20
\]

Dividing both sides by 1.15:

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

Calculating the right side:

\[
t < 17.3913 \quad (\text{approximately})
\]

Combining both inequalities, we have:

\[
15.2174 < t < 17.3913
\]

Therefore, any value of \( t \) within this range would make the total cost fall between $17.50 and $20.

For example, we could choose \( t = 16 \), which is within this range.