Question

Jasmine owns a small manufacturing business. In 2 years, she wants to buy a machine that costs $7,000.00. If Jasmine opens a savings account that earns 7% interest compounded continuously, how much will she have to deposit as principal to have enough money in 2 years to buy the machine?
Round your answer to the nearest cent.

Answer 1

To determine how much Jasmine needs to deposit as principal \( P \) in order to accumulate $7,000 in 2 years with a continuous compounding interest rate of 7%, we can use the formula for continuous compounding:

\[
A = Pe^{rt}
\]

Where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (the initial deposit or investment),
- \( r \) is the annual interest rate (decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

For this situation:
- \( A = 7000 \)
- \( r = 0.07 \)
- \( t = 2 \)

We need to rearrange the formula to solve for \( P \):

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

Now we can plug in the values:

\[
P = \frac{7000}{e^{0.07 \cdot 2}}
\]

First, we calculate \( rt \):

\[
rt = 0.07 \times 2 = 0.14
\]

Next, we calculate \( e^{0.14} \):

Using a calculator or exponentiation:

\[
e^{0.14} \approx 1.150273
\]

Now, we can calculate \( P \):

\[
P = \frac{7000}{1.150273} \approx 6081.63
\]

Thus, the amount Jasmine needs to deposit as principal to accumulate $7,000 in 2 years with continuous compounding at 7% is approximately:

\[
\boxed{6081.63}
\]