How much money needs to be invested now to obtain $3,000 in 10 years if the interest in a savings account is 1.15%, compounded continuously? (Round your answer to the nearest cent.)

To find the amount of money that needs to be invested now to reach a target sum with continuous compounding, we use the formula for continuous compounding:
\[A = P e^{rt}\]
Where:

$A$ is the final amount, which is $\$3,000$.
$P$ is the initial principal (the amount we need to find).
$r$ is the annual interest rate as a decimal, which is $1.15\% = 0.0115$.
$t$ is the time the money is invested in years, which is $10$ years.

Rearrange the Formula and Substitute Values

We need to solve for the principal $P$. Rearranging the formula $A = P e^{rt}$ to isolate $P$, we get:
\[P = \frac{A}{e^{rt}} = A e^{-rt}\]
Substituting the known values:
\[P = 3000 \times e^{-(0.0115 \times 10)}\]
\[P = 3000 \times e^{-0.115}\]

Perform the Calculation and Final Rounding

Calculating the value of $e^{-0.115}$ and multiplying by $3000$:
\[P \approx 3000 \times 0.891366\]
\[P \approx 2674.0984\]
Rounding to the nearest cent, we get $\$2,674.10$.

Examples

Continuous compounding is a fundamental concept in finance, often used to model growth in accounts or investments over time. For instance, if you are planning for a long-term goal like saving for a home down payment or a child's education fund, understanding how much you need to set aside today given an expected growth rate helps in creating realistic financial goals. By using this formula, you can determine exactly how your initial investment grows over time, ensuring your savings strategy stays on track.