Question
Natalie uses a prepaid card to buy movie tickets. The initial value of the card is $180, and each movie ticket costs $12. Let m(x) represent the amount of money, m, in dollars that remains on the card after purchasing x movie tickets.(6 points)
Part A: The function is m(x) =
Part B: Graph the function. (Link to Desmos)
Part C: The constraints are:
and
Part D: For every movie ticket bought, the value of the gift card
by $
Part E: Find the solution of m(x) = 60
x=
Part A: The function is m(x) =
Part B: Graph the function. (Link to Desmos)
Part C: The constraints are:
and
Part D: For every movie ticket bought, the value of the gift card
by $
Part E: Find the solution of m(x) = 60
x=
Answers
GPT-4o mini
Let's go through the parts one by one.
### Part A: The function is
The initial value of the card is $180, and each movie ticket costs $12. If \(x\) represents the number of movie tickets purchased, then the amount of money remaining on the card can be calculated as:
\[
m(x) = 180 - 12x
\]
So, the function is:
\[
m(x) = 180 - 12x
\]
---
### Part B: Graph the function
To graph the function \(m(x) = 180 - 12x\), you would create a graph where the x-axis represents the number of movie tickets bought (x), and the y-axis represents the amount of money remaining on the card (m).
You can use an online graphing calculator like Desmos to plot this function. Simply input the function and view its linear graph.
---
### Part C: The constraints are:
The constraints on \(x\) (number of movie tickets that can be purchased) depend on the initial amount on the card and the cost of each movie ticket.
1. The minimum number of tickets bought is \(0\):
\(x \geq 0\)
2. The maximum number of tickets bought can be found by solving the inequality where the remaining amount \(m(x)\) is greater than or equal to $0:
\[
180 - 12x \geq 0 \implies 180 \geq 12x \implies x \leq 15
\]
So the constraints are:
\[
0 \leq x \leq 15
\]
---
### Part D: For every movie ticket bought, the value of the gift card
The cost of each movie ticket is $12. Therefore, for every movie ticket bought, the value of the gift card:
\[
\text{decreases by } 12 \text{ dollars.}
\]
---
### Part E: Find the solution of \(m(x) = 60\)
To find the number of tickets purchased when the remaining value on the card is $60, set the function equal to 60:
\[
180 - 12x = 60
\]
Now, solve for \(x\):
1. Subtract 180 from both sides:
\[
-12x = 60 - 180
\]
\[
-12x = -120
\]
2. Now divide by -12:
\[
x = \frac{-120}{-12} = 10
\]
So the solution of \(m(x) = 60\) is:
\[
x = 10
\]
---
Let me know if you need any further assistance!
### Part A: The function is
The initial value of the card is $180, and each movie ticket costs $12. If \(x\) represents the number of movie tickets purchased, then the amount of money remaining on the card can be calculated as:
\[
m(x) = 180 - 12x
\]
So, the function is:
\[
m(x) = 180 - 12x
\]
---
### Part B: Graph the function
To graph the function \(m(x) = 180 - 12x\), you would create a graph where the x-axis represents the number of movie tickets bought (x), and the y-axis represents the amount of money remaining on the card (m).
You can use an online graphing calculator like Desmos to plot this function. Simply input the function and view its linear graph.
---
### Part C: The constraints are:
The constraints on \(x\) (number of movie tickets that can be purchased) depend on the initial amount on the card and the cost of each movie ticket.
1. The minimum number of tickets bought is \(0\):
\(x \geq 0\)
2. The maximum number of tickets bought can be found by solving the inequality where the remaining amount \(m(x)\) is greater than or equal to $0:
\[
180 - 12x \geq 0 \implies 180 \geq 12x \implies x \leq 15
\]
So the constraints are:
\[
0 \leq x \leq 15
\]
---
### Part D: For every movie ticket bought, the value of the gift card
The cost of each movie ticket is $12. Therefore, for every movie ticket bought, the value of the gift card:
\[
\text{decreases by } 12 \text{ dollars.}
\]
---
### Part E: Find the solution of \(m(x) = 60\)
To find the number of tickets purchased when the remaining value on the card is $60, set the function equal to 60:
\[
180 - 12x = 60
\]
Now, solve for \(x\):
1. Subtract 180 from both sides:
\[
-12x = 60 - 180
\]
\[
-12x = -120
\]
2. Now divide by -12:
\[
x = \frac{-120}{-12} = 10
\]
So the solution of \(m(x) = 60\) is:
\[
x = 10
\]
---
Let me know if you need any further assistance!