Vinh pays a convenience fee when he reserves movie tickets on his cell phone app. The app shows him the total cost of his purchase for the different number of tickets.

Question

What is the equation that models this linear function? Show your work. Don’t forget to define your variables!

Describe the situation using your vocabulary:

How much is each movie ticket?

What is the interpretation of the y-intercept?

GPT-4o mini · Answer

To create a model for Vinh's total cost when he reserves movie tickets through his cell phone app, we need to define our variables and set up a linear equation.

Definition of Variables:

Let $ x $ be the number of movie tickets Vinh purchases.
Let $ y $ be the total cost of the purchase (including the convenience fee) in dollars.

Suppose:

The price of each movie ticket is $ p $ dollars.
The convenience fee for using the app is $ f $ dollars.

Modeling the Total Cost: The total cost $ y $ can be expressed in terms of the number of tickets $ x $ using the following linear equation:

\[ y = px + f \]

Where:

$ px $ represents the total cost of $ x $ tickets at $ p $ dollars each.
$ f $ represents the convenience fee.

Answering the Questions:

How much is each movie ticket? The price of each movie ticket, denoted as $ p $, is the coefficient of $ x $ in the linear equation. If we know the values for the total cost and the convenience fee, we can determine the value of $ p $ by substituting known values into the equation to solve for $ p $.
What is the interpretation of the y-intercept? In the equation $ y = px + f $, the y-intercept is represented by $ f $, which is the convenience fee. The y-intercept indicates the total cost Vinh incurs when he purchases zero tickets (i.e., $ x = 0 $). In this case, the total cost is equal to the convenience fee itself, as he hasn’t bought any tickets yet. Thus, the y-intercept tells us what Vinh pays just to use the app, regardless of how many tickets he buys.

Summary Formula:

The linear equation that models the situation is:

\[ y = px + f \]

where $ p $ is the price of each movie ticket and $ f $ is the convenience fee.

Answer

A. what is the equation that models this linear function? B. describe the situation using your vocabulary C. how much is each movie ticket D. What is the interpretation of the y-intercept

GPT-4o mini · Answer

To address your questions systematically:

A. What is the equation that models this linear function?

Let:

$ x $ = the number of movie tickets purchased
$ y $ = the total cost of the purchase (in dollars)
$ p $ = the price of each movie ticket (in dollars)
$ f $ = the convenience fee (in dollars)

The equation that models this linear function is:

\[ y = px + f \]

B. Describe the situation using your vocabulary

Vinh uses a cell phone app to reserve movie tickets, which incurs a convenience fee in addition to the cost of the tickets themselves. The app displays the total cost based on how many tickets he wishes to purchase. Therefore, the total cost ($ y $) is determined by the price of each ticket multiplied by the number of tickets purchased ($ px $), plus an additional convenience fee ($ f $) that is charged for using the app.

C. How much is each movie ticket?

The price of each movie ticket is represented by the variable $ p $ in the equation $ y = px + f $. To find $ p $, you would need specific values for $ y $ (the total cost) and $ f $ (the convenience fee). Once those values are known, you can rearrange the equation and solve for $ p $:

\[ p = \frac{y - f}{x} \]

This gives us the price per ticket based on the total cost after accounting for the convenience fee and the number of tickets purchased.

D. What is the interpretation of the y-intercept?

In the equation $ y = px + f $, the y-intercept is $ f $. The interpretation of the y-intercept is as follows:

The y-intercept indicates the total cost when no movie tickets are purchased (i.e., $ x = 0 $). In this case, the total cost $ y $ equals the convenience fee $ f $. This means that if Vinh reserves tickets using the app without buying any tickets, he still incurs the cost of the convenience fee. Thus, the y-intercept represents the fixed cost associated with using the app, independent of the number of tickets purchased.

In summary, to model Vinh's total cost:

The equation is $ y = px + f $.
It encompasses the cost of the tickets and a base fee.
Each movie ticket's price can be determined if you know the total cost and the convenience fee.
The y-intercept reflects the convenience fee paid even when no tickets are bought.

Answer

tickets total cost 2 $32.00 3 $44.00 6 $82.00

GPT-4o mini · Answer

Given the data: - 2 tickets: total cost = $32.00 - 3 tickets: total cost = $44.00 - 6 tickets: total cost = $82.00 We can use this information to determine the price of each movie ticket and the convenience fee. ### A. What is the equation that models this linear function? As established earlier, we can model the total cost using the equation: $$ y = px + f $$ Where $ p $ is the price per ticket and $ f $ is the convenience fee. ### B. Describe the situation using your vocabulary In this scenario, Vinh is purchasing movie tickets through a mobile app that charges a convenience fee in addition to the ticket costs. The total cost includes both the cost for the number of tickets purchased and the app's convenience fee. ### C. How much is each movie ticket? To find $ p $ (the price per ticket) and $ f $ (the convenience fee), we can set up a system of equations based on the provided data points. From the data, we have: 1. For 2 tickets: $$ 2p + f = 32 \tag{1} $$ 2. For 3 tickets: $$ 3p + f = 44 \tag{2} $$ 3. For 6 tickets: $$ 6p + f = 82 \tag{3} $$ Now, we can solve these equations step by step. **Step 1: Subtract equation (1) from equation (2)**: $$ (3p + f) - (2p + f) = 44 - 32 $$ $$ p = 12 \tag{4} $$ **Step 2: Substitute $ p = 12 $ into equation (1) to find $ f $**: $$ 2(12) + f = 32 $$ $$ 24 + f = 32 $$ $$ f = 32 - 24 = 8 \tag{5} $$ ### Summary of Values - Price per ticket, $ p = 12 $ dollars - Convenience fee, $ f = 8 $ dollars ### The Equation Therefore, the equation that models this linear function is: $$ y = 12x + 8 $$ ### D. What is the interpretation of the y-intercept? In the equation $ y = 12x + 8 $, the y-intercept is $ 8 $. **Interpretation**: The y-intercept represents the convenience fee charged if no tickets are purchased. Therefore, if Vinh uses the app but buys 0 tickets, he would still incur a cost of $8, which is the convenience fee charged for making the reservation through the app.