Question

Tasmania uses a gift card to buy video games. The initial value of the gift card is $250 and each game costs $15. V(x) represents the amount of money, V , in dollars that is still left on the card after purchasing,x, games
The function V(x) =. ______
The constraints are ______ and ____
For every game bought the value of the card ____ by ___
Find the solution of
V(x)= 70
x = ___

Answer 1

To start, we can define the function \( V(x) \) to represent the amount of money left on the gift card after purchasing \( x \) video games. Each game costs $15, and the initial value of the gift card is $250.

1. **Function Definition**:
The value left on the gift card after buying \( x \) games can be expressed as:
\[
V(x) = 250 - 15x
\]

2. **Constraints**:
The constraints are:
- \( x \geq 0 \) (you cannot purchase a negative number of games)
- \( V(x) \geq 0 \) (the money left on the card cannot be negative)

To determine the second constraint, we can set \( V(x) \geq 0 \):
\[
250 - 15x \geq 0
\]
This simplifies to:
\[
15x \leq 250 \quad \Rightarrow \quad x \leq \frac{250}{15} \approx 16.67
\]
Since \( x \) must be a whole number, the maximum number of games \( x \) that can be bought is 16. Therefore, the second constraint can be stated as:
- \( x \leq 16 \)

3. **Value Change**:
For every game bought, the value of the card decreases by $15.

4. **Finding the Solution for \( V(x) = 70 \)**:
To find the number of games purchased when the value on the card is $70:
\[
V(x) = 70
\]
Substituting the function into the equation gives:
\[
250 - 15x = 70
\]
Rearranging the equation, we find:
\[
15x = 250 - 70
\]
\[
15x = 180
\]
\[
x = \frac{180}{15} = 12
\]

**Final Answer**:
- The function \( V(x) = 250 - 15x \)
- The constraints are \( x \geq 0 \) and \( x \leq 16 \)
- For every game bought, the value of the card decreases by $15.
- The solution is \( x = 12 \).

Answer 2

Constraint options
0<=y<=16
0<y<250
0<=y<=250
16<=y<=250

Answer 3

Given the context of the problem where \( y \) represents the number of games purchased (let's say \( y \) instead of \( x \) for consistency with your options), we can clarify the constraints based on our previous deductions.

### Constraints Description:

1. We have the condition that the number of games purchased \( y \) cannot be negative:
\[
y \geq 0
\]

2. The maximum number of games that can be purchased is determined by the initial gift card value of $250 and the cost of each game being $15. From earlier calculations, we determined:
\[
15y \leq 250 \quad \Rightarrow \quad y \leq \frac{250}{15} \approx 16.67
\]
Since \( y \) must be a whole number (you cannot purchase a fraction of a game), the maximum number of games that can be purchased is 16. Thus we have:
\[
y \leq 16
\]

### Therefore, the correct constraints are:
\[
0 \leq y \leq 16
\]

Now looking at the possible responses you provided:
1. \( 0 \leq y \leq 16 \) (Correct)
2. \( 0 < y < 250 \) (Incorrect, because \( y \) can be 0 and does not need to be less than 250)
3. \( 0 \leq y \leq 250 \) (Incorrect; \( y \) can only be between 0 and 16)
4. \( 16 \leq y \leq 250 \) (Incorrect; implies \( y \) must be at least 16, which is not true)

### Final Answer for Constraints:
- **Correct constraints**: \( 0 \leq y \leq 16 \)