The value of a collector's 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. \( \$ 578.81 \) \( \$ 607.75 \) \( \$ 1687.50 \) \( \$ 2531.25 \)

To determine the value of the collector's item after 4 years, you can use the formula for exponential growth: \[ A = P(1 + r)^t \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial sum of money, which is \( \$500 \)). - \( r \) is the annual interest rate (decimal) (which is \( 0.05 \) for \( 5\% \)). - \( t \) is the time the money is invested or borrowed for, in years (which is \( 4 \) years). Plugging in the values: \[ A = 500(1 + 0.05)^4 \] Calculating: \[ A = 500(1.21550625) \approx 607.75 \] Thus, the value of the item after 4 years is approximately \( \$607.75 \). --- In the world of collecting, the rarity and condition of an item can make a huge difference in its future value. Collectibles like stamps, coins, or vintage toys often appreciate in value substantially, sometimes outpacing even standard investments! Always remember to keep your collector's items in optimal condition. Factors like humidity, temperature changes, and handling can significantly affect an item's condition—and thus its market value. Use protective cases and store them properly to safeguard your treasures for the future!