Conor is going to the movie theater. A ticket 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

To find the total cost for Conor to buy a movie ticket including the amusement tax, we can break it down as follows:

1. Cost of the ticket: This is simply \( t \) dollars.
2. Amusement tax: The amusement tax is 15% of the ticket price. The tax can be expressed as \( 0.15t \) (which is 15% of \( t \)).

To find the total cost, we need to add the cost of the ticket to the amusement tax:

\[ \text{Total Cost} = \text{Cost of Ticket} + \text{Amusement Tax} \] \[ \text{Total Cost} = t + 0.15t \]

Now we can simplify this expression:

\[ \text{Total Cost} = t(1 + 0.15) \] \[ \text{Total Cost} = t(1.15) \]

Thus, we have two expressions for the total cost:

1. Expression with additions: \( t + 0.15t \)
2. Expression simplified: \( 1.15t \)

Both expressions represent Conor's total cost for the movie ticket including the amusement tax.

Answer 2

can break down the steps more

Answer 3

Sure! Let's break down the steps more clearly:

Step 1: Identify the Cost of the Ticket

• Cost of the ticket: Let \( t \) represent the cost of one movie ticket in dollars.

Step 2: Calculate the Amusement Tax

• Amusement tax rate: The amusement tax is 15%, which can be expressed as 0.15 in decimal form.
• Calculation of the tax: The tax on the ticket price \( t \) can be calculated by multiplying the ticket price by the tax rate: \[ \text{Amusement Tax} = 0.15 \times t \]

Step 3: Find Total Cost

• Total cost is the sum of the ticket price and the amusement tax: \[ \text{Total Cost} = \text{Cost of Ticket} + \text{Amusement Tax} \] Plugging in the known values, we get: \[ \text{Total Cost} = t + (0.15 \times t) \]

Step 4: Simplify the Expression

1. To combine the terms in the total cost, notice that \( t + 0.15t \) can be combined since they both contain \( t \): \[ t + 0.15t = (1 + 0.15)t = 1.15t \]

Summary of Expressions

So, you have two expressions for his total cost:

1. Using addition of the ticket cost and tax: \[ \text{Total Cost} = t + 0.15t \]

2. Using the simplified expression: \[ \text{Total Cost} = 1.15t \]

Both expressions give the same total cost for the ticket including the amusement tax.

Answer 4

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

Answer 5

Let's choose a value for \( t \) that is greater than $12 and less than $13. For this example, we will use \( t = 12.50 \).

Step 1: Use the Simplified Expression

We'll use the simplified expression to calculate Conor's total cost: \[ \text{Total Cost} = 1.15t \]

Step 2: Substitute the Chosen Value

Now, we will substitute \( t = 12.50 \) into the expression: \[ \text{Total Cost} = 1.15 \times 12.50 \]

Step 3: Perform the Calculation

Now, we will multiply: \[ 1.15 \times 12.50 = 14.375 \]

Step 4: Round to the Nearest Penny

To round to the nearest penny, we look at the third decimal place (which is 5 in this case). Since it is 5 or more, we round up: \[ 14.375 \text{ rounds to } 14.38 \]

Final Result

Conor's total cost for the movie ticket, when \( t = 12.50 \), is $14.38.

Answer 6

Now suppose that Connor'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 7

To determine values of \( t \) that would make Conor's total cost fall between $17.50 and $20, we can use the simplified expression for total cost:

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

We want to find \( t \) such that:

\[ 17.50 < 1.15t < 20 \]

Step 1: Divide the Inequalities by 1.15

To isolate \( t \), divide all parts of the inequality by 1.15:

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

Step 2: Perform the Calculations

Now, calculate the lower and upper bounds for \( t \):

Lower Bound: \[ \frac{17.50}{1.15} \approx 15.2174 \]

Upper Bound: \[ \frac{20}{1.15} \approx 17.3913 \]

Step 3: Determine Possible Values for \( t \)

To satisfy the condition \( 17.50 < 1.15t < 20 \), \( t \) must lie between approximately 15.22 and 17.39:

\[ 15.22 < t < 17.39 \]

Conclusion

Any value of \( t \) that falls within this range will result in a total cost that is between $17.50 and $20.

For example:

• A value of \( t = 16 \) would work, since \( 15.22 < 16 < 17.39 \).

Calculating this: \[ \text{Total Cost} = 1.15 \times 16 = 18.40 \] which is indeed greater than $17.50 and less than $20.