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Preview Activity 11.1.1 .
Suppose you invest money in an account that pays 5% interest compounded continuously. If you have an initial investment of \(P\) dollars in the account, then \(A\text{,}\) the amount of money in the account after \(t\) years is given by
\begin{equation*}
A = Pe^{0.05t}.
\end{equation*}
The variables \(P\) and \(t\) are independent of each other, so using functional notation we write
\begin{equation*}
A(P,t) = Pe^{0.05t}.
\end{equation*}
(a) Find the amount of money in the account after 7 years if you originally invest 1000 dollars.
(b) Evaluate
\(A(5000,8)\text{.}\) Write a sentence to explain what this calculation represents.
(c) Now consider only the situation where the amount invested is fixed at 1000 dollars. Calculate the amount of money in the account after
\(t\) years as indicated in the table below. Round payments to the nearest penny.
Duration (in years)
2
3
4
5
6
Amount (dollars)
(d) Now consider the situation where we want to know the amount of money in the account after 10 years given various initial investments. Calculate the amount of money in the account as indicated in the table below. Round payments to the nearest penny.
Initial investment (dollars)
500
1000
5000
7500
10000
Amount (dollars)
(e) Describe as best you can what combinations of initial investments and time will result in an account containing $10,000.
