To determine the correct equation for finding the value [tex]\( y \)[/tex] of the poster after [tex]\( x \)[/tex] years with the information that it starts at [tex]\( \$ 18 \)[/tex] and increases at a rate of [tex]\( 15\% \)[/tex] per year, follow these steps:
1. Identify the initial value and the increase rate:
- The initial value of the poster is [tex]\( \$ 18 \)[/tex].
- The poster increases in value by [tex]\( 15\% \)[/tex] each year, which can be represented as a decimal, [tex]\( 0.15 \)[/tex].
2. Calculate how the value changes each year:
- The value after 1 year is [tex]\( \$ 20.70 \)[/tex], confirming the [tex]\( 15\% \)[/tex] increase from the initial value of [tex]\( \$ 18 \)[/tex].
3. Formulate the general equation:
- The general formula for compound growth is [tex]\( y = \text{initial value} \times (1 + \text{rate of increase})^x \)[/tex].
- Here, the initial value is [tex]\( 18 \)[/tex] and the rate of increase is [tex]\( 0.15 \)[/tex], so the equation becomes:
[tex]\[
y = 18 \times (1 + 0.15)^x
\][/tex]
4. Simplify the equation:
- Simplify the term inside the parentheses: [tex]\( 1 + 0.15 = 1.15 \)[/tex].
[tex]\[
y = 18 \times 1.15^x
\][/tex]
5. Verify using the given values:
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 18 \times 1.15^1 = 18 \times 1.15 = 20.7
\][/tex]
- This matches the given value after 1 year, confirming the validity of our equation.
Thus, the correct equation to find the value [tex]\( y \)[/tex] of the poster after [tex]\( x \)[/tex] years is:
[tex]\[
y = 18 \times 1.15^x
\][/tex]
So, the correct choice is:
[tex]\[
y = 18 (1.15)^x
\][/tex]
