Question

the value of a collectors item is expected to increase exponentially each year. the item is purchased for $500 and its value increases at a rate of 5% per year. find the value of the item after 4 years.
○ $578.81
○ $607.75
○ $1687.50
○ $2531.25

Explanation:

Step1: Recall the exponential growth formula

The formula for exponential growth is \( A = P(1 + r)^t \), where \( A \) is the final amount, \( P \) is the principal amount, \( r \) is the annual growth rate (in decimal), and \( t \) is the time in years.

Step2: Identify the values of \( P \), \( r \), and \( t \)

Given that \( P = 500 \) (the initial purchase price), \( r = 0.05 \) (since 5% = 0.05), and \( t = 4 \) (the number of years).

Step3: Substitute the values into the formula

Substitute \( P = 500 \), \( r = 0.05 \), and \( t = 4 \) into the formula \( A = P(1 + r)^t \):

$$ A = 500(1 + 0.05)^4 $$

Step4: Calculate \( (1 + 0.05)^4 \)

First, calculate \( 1 + 0.05 = 1.05 \). Then, calculate \( 1.05^4 \).
\( 1.05^4 = 1.05\times1.05\times1.05\times1.05 = 1.21550625 \)

Step5: Calculate the final amount \( A \)

Multiply \( 500 \) by \( 1.21550625 \):

$$ A = 500\times1.21550625 = 607.753125 $$

Rounding to two decimal places, we get \( A \approx 607.75 \)

Answer:

\$607.75