Question 1

Question

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

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

Let's break down the problem step by step.

Part a)

To express Conor's total cost in terms of \( t \), we need to account for the cost of the ticket and the amusement tax.

1. The cost of the ticket is \( t \).
2. The amusement tax is 15% of the ticket price, which can be expressed as \( 0.15t \).

Now, the total cost \( C \) can be expressed as:

Expression 1: \[ C = t + 0.15t = 1.15t \]

Alternatively, you could express the tax added separately:

Expression 2: \[ C = t + (0.15t) = t + 0.15t \]

Both expressions will yield the same result.

Part b)

Let's choose a value for \( t \) within the range stated (greater than $12 and less than $13).

Let's pick \( t = 12.50 \).

Using Expression 1:

\[ C = 1.15t \] Substituting \( t \):

\[ C = 1.15(12.50) = 14.375 \]

Rounded to the nearest penny, Conor’s total cost is:

\[ C \approx 14.38 \]

Part c)

We need to find a value of \( t \) such that Conor's total cost \( C \) is greater than $17.50 and less than $20.

Using Expression 1 again:

Setting up inequalities:

\[ 17.50 < 1.15t < 20 \]

Now we will divide the entire inequality by 1.15 to isolate \( t \):

1. First inequality: \[ \frac{17.50}{1.15} < t \] Calculating \( \frac{17.50}{1.15} \): \[ t > 15.217 \text{ (approximately)} \]

2. Second inequality: \[ t < \frac{20}{1.15} \] Calculating \( \frac{20}{1.15} \): \[ t < 17.391 \text{ (approximately)} \]

Final range for \( t \):

So, the values of \( t \) that satisfy the condition are: \[ 15.217 < t < 17.391 \]

You can choose any value within this range. For example, \( t = 16 \) works.

Example Check:

If \( t = 16 \): \[ C = 1.15(16) = 18.4 \]

And indeed, \( 18.4 \) is between \( 17.50 \) and \( 20 \).

Answer 2

I missed all the answers.

Answer 3

Let's go through the answers step by step clearly.

Part a)

We need to write two expressions for Conor's total cost \( C \) when he buys a ticket to a movie priced at \( t \) dollars and a 15% amusement tax on each ticket.

1. Expression for total cost:
   The cost of the ticket is \( t \). The amusement tax on the ticket is \( 0.15t \). Therefore:
   • Expression 1: \[ C = t + 0.15t = 1.15t \]
   • Expression 2: \[ C = t + 0.15t = t + (0.15t) \]

Part b)

Next, choose a value for \( t \) that is greater than $12 and less than $13. Let's choose \( t = 12.50 \).

Using Expression 1: \[ C = 1.15t \] Substituting \( t = 12.50 \): \[ C = 1.15(12.50) = 14.375 \]

Rounded to the nearest penny, Conor’s total cost is: \[ C \approx 14.38 \]

Part c)

Now, we need to find a value of \( t \) such that Conor's total cost \( C \) is greater than $17.50 and less than $20. We can use Expression 1:

We set up the inequalities: \[ 17.50 < 1.15t < 20 \]

Dividing the entire inequality by 1.15 to find the range for \( t \):

1. For the left side: \[ \frac{17.50}{1.15} < t \] Evaluating: \[ t > 15.217 \text{ (approximately)} \]

2. For the right side: \[ t < \frac{20}{1.15} \] Evaluating: \[ t < 17.391 \text{ (approximately)} \]

So, the range for \( t \) is: \[ 15.217 < t < 17.391 \]

You can pick any value for \( t \) within this range. For example, \( t = 16 \) works.

Example Check:

If \( t = 16 \): \[ C = 1.15(16) = 18.4 \] This means Conor's total cost is \( 18.4 \), which is between \( 17.50 \) and \( 20 \).

Summary of Answers:

• Expression for total cost: \( C = 1.15t \) or \( C = t + 0.15t \)
• Total cost at \( t = 12.50 \): \( 14.38 \)
• Range for \( t \) to satisfy total cost between \( 17.50 \) and \( 20 \): \( 15.217 < t < 17.391 \) (for example, pick \( t = 16 \)).