Questions: How much would you have to deposit in an account with a 9% interest rate, compounded continuously, to have 1500 in your account 5 years later? P=[?] Round to the nearest cent.

How much would you have to deposit in an account with a 9% interest rate, compounded continuously, to have 1500 in your account 5 years later?
P=[?]

Round to the nearest cent.

Transcript text: How much would you have to deposit in an account with a $9 \%$ interest rate, compounded continuously, to have $\$ 1500$ in your account 5 years later? \[ \mathrm{P}=\$[?] \] Round to the nearest cent.

Solution

Solution Steps

Step 1: Identify the Variables

We are given the following values:

$ A = 1500 $ (the amount of money desired after 5 years)
$ r = 0.09 $ (the annual interest rate)
$ t = 5 $ (the time in years)

Step 2: Use the Continuous Compounding Formula

We will use the formula for continuous compounding interest:

\[ A = P e^{rt} \]

To find the principal amount $ P $, we rearrange the formula:

\[ P = \frac{A}{e^{rt}} \]

Step 3: Substitute the Values

Substituting the known values into the rearranged formula:

\[ P = \frac{1500}{e^{0.09 \cdot 5}} \]

Step 4: Calculate the Exponential Component

Calculate $ e^{0.09 \cdot 5} $:

\[ e^{0.45} \]

Step 5: Compute the Principal Amount

Now, compute $ P $:

\[ P = \frac{1500}{e^{0.45}} \]

Step 6: Round the Result

Finally, round $ P $ to the nearest cent to find the amount that needs to be deposited.

\[ P \approx 956.44 \]

Final Answer

$\boxed{P \approx 956.44}$

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