Question
Bobby and Emilia deposit $600.00 into a savings account which earns 3% interest compounded continuously. They want to use the money in the account to go on a trip in 1 year. How much will they be able to spend?
Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.
Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.
Answers
Answered by
GPT-4o mini
To find the amount that Bobby and Emilia will have in their savings account after one year with continuous compounding, we can use the formula:
\[
A = Pe^{rt}
\]
Where:
- \(A\) is the amount of money accumulated after time \(t\),
- \(P\) is the principal amount (the initial deposit),
- \(e\) is approximately 2.71828,
- \(r\) is the annual interest rate (as a decimal),
- \(t\) is the time in years.
Given:
- \(P = 600.00\)
- \(r = 0.03\) (3% as a decimal)
- \(t = 1\)
Now we can plug in the values:
\[
A = 600.00 \cdot e^{0.03 \cdot 1}
\]
First, calculate \(0.03 \cdot 1\):
\[
0.03 \cdot 1 = 0.03
\]
Now calculate \(e^{0.03}\):
Using the approximation \(e \approx 2.71828\):
\[
e^{0.03} \approx 2.71828^{0.03} \approx 1.03045453395
\]
Now, multiply the principal by \(e^{0.03}\):
\[
A \approx 600.00 \cdot 1.03045453395
\]
Calculating the multiplication:
\[
A \approx 618.27272037
\]
Now round to the nearest cent:
\[
A \approx 618.27
\]
Thus, Bobby and Emilia will be able to spend approximately **$618.27**.
\[
A = Pe^{rt}
\]
Where:
- \(A\) is the amount of money accumulated after time \(t\),
- \(P\) is the principal amount (the initial deposit),
- \(e\) is approximately 2.71828,
- \(r\) is the annual interest rate (as a decimal),
- \(t\) is the time in years.
Given:
- \(P = 600.00\)
- \(r = 0.03\) (3% as a decimal)
- \(t = 1\)
Now we can plug in the values:
\[
A = 600.00 \cdot e^{0.03 \cdot 1}
\]
First, calculate \(0.03 \cdot 1\):
\[
0.03 \cdot 1 = 0.03
\]
Now calculate \(e^{0.03}\):
Using the approximation \(e \approx 2.71828\):
\[
e^{0.03} \approx 2.71828^{0.03} \approx 1.03045453395
\]
Now, multiply the principal by \(e^{0.03}\):
\[
A \approx 600.00 \cdot 1.03045453395
\]
Calculating the multiplication:
\[
A \approx 618.27272037
\]
Now round to the nearest cent:
\[
A \approx 618.27
\]
Thus, Bobby and Emilia will be able to spend approximately **$618.27**.
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