Question
In 1 year, Bernie wants to buy a bicycle that costs $400.00. If he opens a savings account that earns 3% interest compounded continuously, how much will he have to deposit as principal to have enough money in 1 year to buy the bike?
Round your answer to the nearest cent.
$
Round your answer to the nearest cent.
$
Answers
GPT-4o mini
To determine how much Bernie needs to deposit as principal to have $400 in 1 year with an account that earns 3% interest compounded continuously, we can use the formula for continuously compounded interest:
\[
A = Pe^{rt}
\]
Where:
- \( A \) is the amount of money accumulated after time \( t \) (in this case, $400),
- \( P \) is the principal amount (the initial deposit),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time the money is invested for (in years),
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
We can rearrange the formula to solve for \( P \):
\[
P = \frac{A}{e^{rt}}
\]
Now, substituting the known values into the equation:
- \( A = 400 \)
- \( r = 0.03 \) (3% as a decimal)
- \( t = 1 \)
Plugging these values into the formula gives:
\[
P = \frac{400}{e^{0.03 \times 1}} = \frac{400}{e^{0.03}}
\]
First, we need to calculate \( e^{0.03} \):
\[
e^{0.03} \approx 1.0304545339 \quad (\text{using a calculator})
\]
Now substitute this value back into the equation:
\[
P = \frac{400}{1.0304545339} \approx 387.232
\]
Rounding to the nearest cent:
\[
P \approx 387.23
\]
Thus, Bernie will need to deposit approximately \( \boxed{387.23} \) dollars to have enough money to buy the bicycle in one year.
\[
A = Pe^{rt}
\]
Where:
- \( A \) is the amount of money accumulated after time \( t \) (in this case, $400),
- \( P \) is the principal amount (the initial deposit),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time the money is invested for (in years),
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
We can rearrange the formula to solve for \( P \):
\[
P = \frac{A}{e^{rt}}
\]
Now, substituting the known values into the equation:
- \( A = 400 \)
- \( r = 0.03 \) (3% as a decimal)
- \( t = 1 \)
Plugging these values into the formula gives:
\[
P = \frac{400}{e^{0.03 \times 1}} = \frac{400}{e^{0.03}}
\]
First, we need to calculate \( e^{0.03} \):
\[
e^{0.03} \approx 1.0304545339 \quad (\text{using a calculator})
\]
Now substitute this value back into the equation:
\[
P = \frac{400}{1.0304545339} \approx 387.232
\]
Rounding to the nearest cent:
\[
P \approx 387.23
\]
Thus, Bernie will need to deposit approximately \( \boxed{387.23} \) dollars to have enough money to buy the bicycle in one year.