Question © Video Examples A quantity with an initial value of 450 grows exponentially at a rate of \( 0.4 \% \) every 10 years. What is the value of the quantity after 96 years, to the nearest hundredth?

To solve this exponential growth problem, we can use the formula \( A = P(1 + r)^t \), where \( A \) is the amount after time \( t \), \( P \) is the initial amount, \( r \) is the growth rate per period, and \( t \) is the number of periods. Here, \( P = 450 \), the growth rate \( r = 0.004 \) (converted from percentage), and \( t = \frac{96}{10} = 9.6 \) (since the rate is per 10 years). Therefore, we calculate: \[ A = 450(1 + 0.004)^{9.6} \] \[ A \approx 450(1.004)^{9.6} \approx 450 \times 1.03950 \approx 467.78 \] So the value of the quantity after 96 years, rounded to the nearest hundredth, is approximately \( 467.78 \, g \). Excited to see how exponential growth manifests, did you know that this concept isn't just theoretical? It's applied in real-life scenarios like population growth, money investments, and even determining the spread of diseases! It’s fascinating how a tiny percentage increment can lead to substantial changes over time, demonstrating the butterfly effect in financial planning and environmental considerations. Ever tangled with exponential growth calculations before? A common mistake is to confuse the period of growth with the total time! Remember, the rate given is typically per a specific interval, so always check your time units before diving into calculations. Additionally, not converting the percentage into a decimal form can lead to significant miscalculations—just a little decimal place can make a whole world of difference!