To determine which equation accurately describes the value [tex]\( y \)[/tex] of a limited-edition poster after [tex]\( x \)[/tex] years, knowing it starts with an initial value of \[tex]$18 and increases by 15% per year, we should evaluate each given equation correctly.
### Step-by-Step Solution:
1. Initial Conditions:
- Initial value \( \left( V_0 \right) = \$[/tex]18 \)
- Annual increase rate [tex]\( r = 15\% = 0.15 \)[/tex]
2. Value after 1 year:
- Using the initial value and increase rate, after 1 year, the value would be calculated as:
[tex]\[
V_1 = V_0 \times (1 + r)
\][/tex]
- Plugging in the values, we have:
[tex]\[
V_1 = 18 \times (1 + 0.15) = 18 \times 1.15
\][/tex]
- This results in:
[tex]\[
V_1 = 20.70
\][/tex]
3. Formulating the Equation:
- To generalize the calculation for any number of years [tex]\( x \)[/tex], we need a formula involving the initial value and the growth rate compounded over [tex]\( x \)[/tex] years.
- The correct form of the equation would be [tex]\( y = 18 \times (1.15)^x \)[/tex].
4. Evaluating the Options:
- Option 1: [tex]\( y=18(1.15)^x \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 18 \times (1.15)^1 = 18 \times 1.15 = 20.70
\][/tex]
- This matches the given value after 1 year, validating this form of the equation.
- Option 2: [tex]\( y=18(0.15)^x \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 18 \times (0.15)^1 = 18 \times 0.15 = 2.70
\][/tex]
- This value is significantly lower than \[tex]$20.70, hence this option is incorrect.
- Option 3: \( y=20.7(1.15)^x \)
- When \( x = 1 \):
\[
y = 20.7 \times (1.15)^1 = 20.7 \times 1.15 = 23.805
\]
- This value is higher than the given value of \$[/tex]20.70 after 1 year, so this option is also incorrect.
- Option 4: [tex]\( y=20.7(0.15)^x \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 20.7 \times (0.15)^1 = 20.7 \times 0.15 = 3.105
\][/tex]
- This value is also incorrect as it falls short of \[tex]$20.70 after 1 year.
### Conclusion:
The correct equation to determine the value \( y \) of the limited-edition poster after \( x \) years, given an initial value of \$[/tex]18 and an annual increase of 15%, is:
[tex]\[
\boxed{y = 18(1.15)^x}
\][/tex]
