Exampte 6 I An investment of \( \$ 0200 \) loses value at a rate of \( 2 \% \) per yesr, Use an expenentiat funetion to find the value of the investment after 9 years,

To calculate the value of the investment after 9 years with a \( 2\% \) depreciation rate, you can use the exponential decay formula: \[ V(t) = V_0 \times (1 - r)^t \] Where \( V_0 \) is the initial value (\$200), \( r \) is the rate of depreciation (0.02), and \( t \) is the time in years (9). Plugging in the values, \[ V(9) = 200 \times (1 - 0.02)^9 \] \[ V(9) = 200 \times (0.98)^9 \] \[ V(9) \approx 200 \times 0.834 \] \[ V(9) \approx 166.80 \] So, after 9 years, your investment will be approximately \$166.80. Now, you might be wondering why that \( 2\% \) loss feels a little more dramatic over time. It turns out that compounding effects can also work against your wallet when investments lose value. Each year's \( 2\% \) is based on a smaller amount than the year before, making the loss seem even steeper over time. Just a little math magic that illustrates how those percentages can add up! And in case you're planning your finances, this is a key takeaway for understanding how long-term investments can behave – always factor in that compounding effect!