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.)
A. $y=18(1.15)^x$
B. $y=18(0.15)^x$
C. $y=20.7(1.15)^x$
D. $y=20.7(0.15)^x
Answer (2)
The equation that represents the value of the limited-edition poster after x years is given by y = 18 ( 1.15 ) x . This equation arises from the initial value of 18 an d a g ro wt h r a t eo f 15 y=18(1.15)^x$.
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The problem describes an exponential growth scenario where a poster's value increases by a fixed percentage each year.
The general formula for exponential growth is y = a ( 1 + r ) x , where a is the initial value, r is the growth rate, and x is the number of years.
Substituting the given values, the initial value a = 18 and the growth rate r = 0.15 into the formula.
The equation representing the poster's value after x years is y = 18 ( 1.15 ) x .
Explanation
Understanding the Problem Let's analyze the problem. We are given that a limited-edition poster has an initial value of $18 and increases in value each year by 15%. We need to find an equation that represents the value, y , of the poster after x years.
General Exponential Growth Equation The general form of an exponential growth equation is given by:
y = a ( 1 + r ) x
where:
y is the value after x years,
a is the initial value,
r is the growth rate (as a decimal),
x is the number of years.
Substituting the Values In this problem, we have:
Initial value, a = 18
Growth rate, r = 15% = 0.15
Substituting these values into the general equation, we get:
y = 18 ( 1 + 0.15 ) x
y = 18 ( 1.15 ) x
The Equation Therefore, the equation that represents the value, y , of the poster after x years is:
y = 18 ( 1.15 ) x
Examples
Exponential growth is a mathematical transformation that increases without bound. For example, if you invest $100 in a savings account that yields 5% interest annually, the value of your investment grows exponentially. After 10 years, your investment would be worth approximately $162.89, and after 20 years, it would be worth approximately $265.33. This principle is widely used in finance, economics, and even biology to model population growth or the spread of diseases.
