Question
Jordan uses a prepaid card to buy movie tickets. The initial value of the card is $240, and each movie ticket costs $15. 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=
Answers
GPT-4o mini
### Part A: The function is
To find the function \( m(x) \), which represents the amount of money remaining on the prepaid card after purchasing \( x \) movie tickets, we can use the initial value of the card and the cost of each ticket.
The initial value of the card is $240. Each ticket costs $15. Therefore, after purchasing \( x \) movie tickets, the function would be:
\[
m(x) = 240 - 15x
\]
### Part B: Graph the function.
You can use this equation \( m(x) = 240 - 15x \) and graph it on Desmos or any graphing tool. The graph will be a straight line with a negative slope, crossing the y-axis at 240 and the x-axis when \( m(x) = 0 \).
### Part C: The constraints are:
Since \( m(x) \) represents the money left on the card, it must be non-negative. Therefore, the constraints are:
- \( m(x) \geq 0 \)
Solving \( 240 - 15x \geq 0 \):
\[
240 \geq 15x
\]
Dividing both sides by 15 gives:
\[
16 \geq x
\]
This means:
\[
0 \leq x \leq 16
\]
### Part D: For every movie ticket bought, the value of the gift card decreases by:
The value of the gift card decreases by the price of one movie ticket for every ticket bought. Since each ticket costs $15, we conclude that:
**For every movie ticket bought, the value of the gift card decreases by $15.**
### Part E: Find the solution of \( m(x) = 60 \)
To find the value of \( x \) when \( m(x) = 60 \):
Set the function equal to 60:
\[
240 - 15x = 60
\]
Subtract 240 from both sides:
\[
-15x = 60 - 240
\]
\[
-15x = -180
\]
Dividing both sides by -15 gives:
\[
x = 12
\]
### Final Answers:
- **Part A:** \( m(x) = 240 - 15x \)
- **Part C:** Constraints are \( 0 \leq x \leq 16 \)
- **Part D:** The value of the gift card decreases by $15 for every movie ticket bought.
- **Part E:** \( x = 12 \)
To find the function \( m(x) \), which represents the amount of money remaining on the prepaid card after purchasing \( x \) movie tickets, we can use the initial value of the card and the cost of each ticket.
The initial value of the card is $240. Each ticket costs $15. Therefore, after purchasing \( x \) movie tickets, the function would be:
\[
m(x) = 240 - 15x
\]
### Part B: Graph the function.
You can use this equation \( m(x) = 240 - 15x \) and graph it on Desmos or any graphing tool. The graph will be a straight line with a negative slope, crossing the y-axis at 240 and the x-axis when \( m(x) = 0 \).
### Part C: The constraints are:
Since \( m(x) \) represents the money left on the card, it must be non-negative. Therefore, the constraints are:
- \( m(x) \geq 0 \)
Solving \( 240 - 15x \geq 0 \):
\[
240 \geq 15x
\]
Dividing both sides by 15 gives:
\[
16 \geq x
\]
This means:
\[
0 \leq x \leq 16
\]
### Part D: For every movie ticket bought, the value of the gift card decreases by:
The value of the gift card decreases by the price of one movie ticket for every ticket bought. Since each ticket costs $15, we conclude that:
**For every movie ticket bought, the value of the gift card decreases by $15.**
### Part E: Find the solution of \( m(x) = 60 \)
To find the value of \( x \) when \( m(x) = 60 \):
Set the function equal to 60:
\[
240 - 15x = 60
\]
Subtract 240 from both sides:
\[
-15x = 60 - 240
\]
\[
-15x = -180
\]
Dividing both sides by -15 gives:
\[
x = 12
\]
### Final Answers:
- **Part A:** \( m(x) = 240 - 15x \)
- **Part C:** Constraints are \( 0 \leq x \leq 16 \)
- **Part D:** The value of the gift card decreases by $15 for every movie ticket bought.
- **Part E:** \( x = 12 \)