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 problem describes an exponential growth scenario.
The general form of an exponential growth equation is y = a ( 1 + r ) x .
Substitute the initial value a = 18 and growth rate r = 0.15 into the equation.
The equation representing the value of the poster after x years is y = 18 ( 1.15 ) x .
Explanation
Understanding the Problem We are given that a limited-edition poster has an initial value of $18 and increases in value by 15% each year. We want to find an equation that models 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 exponential growth equation, we get: y = 18 ( 1 + 0.15 ) x y = 18 ( 1.15 ) x
Verification To verify this equation, we can check the value after 1 year: y = 18 ( 1.15 ) 1 = 18 × 1.15 = 20.7 This matches the given information that the poster is worth $20.70 after 1 year.
Final Equation Therefore, the equation that represents the value, y , of the poster after x years is: y = 18 ( 1.15 ) x
Examples
Imagine you bought a rare comic book for $18. If its value increases by 15% each year, this equation helps you predict its future worth. For example, after 5 years, its value would be approximately $36.21. Understanding exponential growth is useful in many real-life scenarios, such as predicting investments, population growth, or even the spread of information.
The equation that represents the value of the poster after x years is y = 18 ( 1.15 ) x . This formula uses the initial value of the poster and a growth rate of 15% per year. The correct answer is option A.
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