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) =
Responses:
12x-180
180-12x
12x+180
180+12x
Part C: The constraints are___
and ____
Responses:
0<x<15
0≤x≤12
0≤x≤15
Responses:
15≤y≤180
0≤y≤15
0<y<180
0≤y≤180
Part D: For every movie ticket bought, the value of the gift card by ____ $ ___
Responses:
Decreases
Increases
Responses:
1
12
15
180
Part E: Find the solution of m(x) = 60
x=
Responses:
12
180
540
10
Part A: The function is m(x) =
Responses:
12x-180
180-12x
12x+180
180+12x
Part C: The constraints are___
and ____
Responses:
0<x<15
0≤x≤12
0≤x≤15
Responses:
15≤y≤180
0≤y≤15
0<y<180
0≤y≤180
Part D: For every movie ticket bought, the value of the gift card by ____ $ ___
Responses:
Decreases
Increases
Responses:
1
12
15
180
Part E: Find the solution of m(x) = 60
x=
Responses:
12
180
540
10
Answers
GPT-4o mini
Let's address each part of the problem step by step.
### Part A: The function m(x)
The initial value of the card is $180, and each movie ticket costs $12. After purchasing x movie tickets, the amount left on the card can be expressed as:
\[
m(x) = \text{initial value} - \text{cost per ticket} \times \text{number of tickets}
\]
So,
\[
m(x) = 180 - 12x
\]
Thus, the correct response is:
**180 - 12x**
### Part B: The constraints
To find the constraints for the number of movie tickets (x) Natalie can buy:
- Since each movie ticket costs $12, the maximum number of tickets she can buy with $180 is \( \frac{180}{12} = 15 \).
- Therefore, x must be non-negative (0) and less than or equal to 15.
So the constraints are:
**0 ≤ x ≤ 15**
For the amount of money remaining (y):
- Since y represents the remaining amount on the card, it must be between 0 and 180.
- The minimum value of y is $0, which occurs when she buys 15 tickets ($180 - $180 = $0).
- The maximum value of y is $180, which occurs when no tickets are bought.
Thus, the correct response is:
**0 ≤ y ≤ 180**
### Part D: Change in value of the card
For every movie ticket bought, the value of the gift card decreases by the cost of one movie ticket, which is $12.
So, the correct response is:
**Decreases**
and
**12**
### Part E: Solve m(x) = 60
We need to solve the equation:
\[
180 - 12x = 60
\]
Subtract 180 from both sides:
\[
-12x = 60 - 180
\]
\[
-12x = -120
\]
Now, divide both sides by -12:
\[
x = 10
\]
Thus, the correct response is:
**10**
In summary:
- Part A: **180 - 12x**
- Part B: **0 ≤ x ≤ 15** and **0 ≤ y ≤ 180**
- Part D: **Decreases** and **12**
- Part E: **10**
### Part A: The function m(x)
The initial value of the card is $180, and each movie ticket costs $12. After purchasing x movie tickets, the amount left on the card can be expressed as:
\[
m(x) = \text{initial value} - \text{cost per ticket} \times \text{number of tickets}
\]
So,
\[
m(x) = 180 - 12x
\]
Thus, the correct response is:
**180 - 12x**
### Part B: The constraints
To find the constraints for the number of movie tickets (x) Natalie can buy:
- Since each movie ticket costs $12, the maximum number of tickets she can buy with $180 is \( \frac{180}{12} = 15 \).
- Therefore, x must be non-negative (0) and less than or equal to 15.
So the constraints are:
**0 ≤ x ≤ 15**
For the amount of money remaining (y):
- Since y represents the remaining amount on the card, it must be between 0 and 180.
- The minimum value of y is $0, which occurs when she buys 15 tickets ($180 - $180 = $0).
- The maximum value of y is $180, which occurs when no tickets are bought.
Thus, the correct response is:
**0 ≤ y ≤ 180**
### Part D: Change in value of the card
For every movie ticket bought, the value of the gift card decreases by the cost of one movie ticket, which is $12.
So, the correct response is:
**Decreases**
and
**12**
### Part E: Solve m(x) = 60
We need to solve the equation:
\[
180 - 12x = 60
\]
Subtract 180 from both sides:
\[
-12x = 60 - 180
\]
\[
-12x = -120
\]
Now, divide both sides by -12:
\[
x = 10
\]
Thus, the correct response is:
**10**
In summary:
- Part A: **180 - 12x**
- Part B: **0 ≤ x ≤ 15** and **0 ≤ y ≤ 180**
- Part D: **Decreases** and **12**
- Part E: **10**