Question

a limited-edition poster increases in value each year with an initial value of $18. after 1 year and an increase of 15% per year, the poster is worth $20.70. which equation can be used to find the value, y, after x years? (round money values to the nearest penny.)

\\(y = 18(1.15)^x\\)
\\(y = 18(0.15)^x\\)
\\(y = 20.7(1.15)^x\\)
\\(y = 20.7(0.15)^x\\)

Explanation:

Identify the initial value

The initial value represents the starting amount when \(x = 0\).
The problem states the initial value is \$18.
Let \(a = 18\).

Determine the growth factor

The value increases by \(15\%\) each year.
The growth rate is \(r = 0.15\).
The growth factor is \(1 + r = 1 + 0.15 = 1.15\).

Formulate the exponential equation

The general exponential growth model is:

$$y = a(1 + r)^x$$

Substitute \(a = 18\) and \(1 + r = 1.15\):

$$y = 18(1.15)^x$$

Verify with the given data point

After \(x = 1\) year, the value is:

$$y = 18(1.15)^1 = 20.70$$

This matches the given value of \$20.70.

Answer:

• (A) \(y = 18(1.15)^x\) (Correct answer)
• (B) \(y = 18(0.15)^x\)
• (C) \(y = 20.7(1.15)^x\)
• (D) \(y = 20.7(0.15)^x\)