Questions: Amelia invested 19,000 in the stock market five years ago and has recorded the value of the investment annually since then, as shown in the table. Year, x Value, f(x) 0 19,000 1 19,400 2 20,100 3 21,200 4 22,400 5 23,800 Use the exponential function from the previous step to estimate the value of the investment after one more year. Round your answer to the nearest 100.

Amelia invested 19,000 in the stock market five years ago and has recorded the value of the investment annually since then, as shown in the table. Year, x Value, f(x) 0 19,000 1 19,400 2 20,100 3 21,200 4 22,400 5 23,800 Use the exponential function from the previous step to estimate the value of the investment after one more year. Round your answer to the nearest 100.
Transcript text: Amelia invested $\$ 19,000$ in the stock market five years ago and has recorded the value of the investment annually since then, as shown in the table. \begin{tabular}{|c|c|} \hline Year, $x$ & Value, $f(x)$ \\ \hline 0 & $\$ 19,000$ \\ \hline 1 & $\$ 19,400$ \\ \hline 2 & $\$ 20,100$ \\ \hline 3 & $\$ 21,200$ \\ \hline 4 & $\$ 22,400$ \\ \hline 5 & $\$ 23,800$ \\ \hline \end{tabular} Step 2 of 2: Use the exponential function from the previous step to estimate the value of the investment after one more year. Round your answer to the nearest $\$ 100$.
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Solution

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Solution Steps

Step 1: Find the exponential function

Let the exponential function be $f(x) = ab^x$. Since $f(0) = 19000$, we have $ab^0 = 19000$, so $a = 19000$. Now we have $f(x) = 19000b^x$. Using the point $(1, 19400)$, we get $19400 = 19000b^1$, which implies $b = 19400/19000 = 1.02105$. Thus, the exponential function is $f(x) = 19000(1.02105)^x$.

Step 2: Estimate the value after one more year

We want to find the value of the investment after one more year, which is at $x=6$. $f(6) = 19000(1.02105)^6 \approx 19000(1.13267) \approx 21520.73$.

Final Answer The final answer is $\boxed{21500}$

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