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Answer :
Sure! To find out how many years it will take for Susie's money to double with continuous compounding, we can use the formula for continuous compound interest:
[tex]\[ A = P \times e^{(rt)} \][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount of money.
- [tex]\( P \)[/tex] is the initial principal (the amount of money initially invested).
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( r \)[/tex] is the annual interest rate.
- [tex]\( t \)[/tex] is the time in years.
In Susie's case:
- The initial investment ([tex]\( P \)[/tex]) is [tex]$500.
- She wants it to double, so the final amount (\( A \)) will be $[/tex]1000.
- The annual interest rate ([tex]\( r \)[/tex]) is 5%, which can be expressed as 0.05 in decimal form.
We need to solve for [tex]\( t \)[/tex], the time it takes for the investment to double. The formula rearranges to:
[tex]\[ t = \frac{\ln(\frac{A}{P})}{r} \][/tex]
Steps to solve:
1. Substitute [tex]\( A = 1000 \)[/tex], [tex]\( P = 500 \)[/tex], and [tex]\( r = 0.05 \)[/tex] into the formula.
2. Calculate the ratio [tex]\( \frac{A}{P} \)[/tex]:
[tex]\[
\frac{1000}{500} = 2
\][/tex]
3. Find the natural logarithm of 2:
[tex]\[
\ln(2) \approx 0.6931
\][/tex]
4. Divide the natural logarithm of 2 by the interest rate:
[tex]\[
\frac{0.6931}{0.05} \approx 13.86
\][/tex]
Therefore, it will take approximately 13.86 years for Susie's initial investment to double with continuous compounding at an annual interest rate of 5%.
[tex]\[ A = P \times e^{(rt)} \][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount of money.
- [tex]\( P \)[/tex] is the initial principal (the amount of money initially invested).
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( r \)[/tex] is the annual interest rate.
- [tex]\( t \)[/tex] is the time in years.
In Susie's case:
- The initial investment ([tex]\( P \)[/tex]) is [tex]$500.
- She wants it to double, so the final amount (\( A \)) will be $[/tex]1000.
- The annual interest rate ([tex]\( r \)[/tex]) is 5%, which can be expressed as 0.05 in decimal form.
We need to solve for [tex]\( t \)[/tex], the time it takes for the investment to double. The formula rearranges to:
[tex]\[ t = \frac{\ln(\frac{A}{P})}{r} \][/tex]
Steps to solve:
1. Substitute [tex]\( A = 1000 \)[/tex], [tex]\( P = 500 \)[/tex], and [tex]\( r = 0.05 \)[/tex] into the formula.
2. Calculate the ratio [tex]\( \frac{A}{P} \)[/tex]:
[tex]\[
\frac{1000}{500} = 2
\][/tex]
3. Find the natural logarithm of 2:
[tex]\[
\ln(2) \approx 0.6931
\][/tex]
4. Divide the natural logarithm of 2 by the interest rate:
[tex]\[
\frac{0.6931}{0.05} \approx 13.86
\][/tex]
Therefore, it will take approximately 13.86 years for Susie's initial investment to double with continuous compounding at an annual interest rate of 5%.
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