To solve this compound interest problem where $7000 is invested in an account with an interest rate of 2.5%, compounded continuously, we need to determine the time required for the amount to double and triple.
We start with the continuous compound interest formula:
[tex]\[ A = P \cdot e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal).
- [tex]\( t \)[/tex] is the time the money is invested for in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
### Part (a) Find the time required for the amount to double
To find the time required for the amount to double, we set [tex]\( A = 2P \)[/tex].
1. Start with the equation [tex]\( 2P = P \cdot e^{rt} \)[/tex].
2. Divide both sides by [tex]\( P \)[/tex]: [tex]\( 2 = e^{rt} \)[/tex].
3. Take the natural logarithm of both sides: [tex]\( \ln(2) = rt \)[/tex].
4. Solve for [tex]\( t \)[/tex]: [tex]\( t = \frac{\ln(2)}{r} \)[/tex].
Given [tex]\( r = 2.5\% = 0.025 \)[/tex]:
[tex]\[ t = \frac{\ln(2)}{0.025} \][/tex]
Using the values:
[tex]\[ \ln(2) \approx 0.693 \][/tex]
[tex]\[ t \approx \frac{0.693}{0.025} \][/tex]
[tex]\[ t \approx 27.73 \][/tex]
Hence, the time required for the amount to double is approximately 27.73 years.
### Part (b) Find the time required for the amount to triple
To find the time required for the amount to triple, we set [tex]\( A = 3P \)[/tex].
1. Start with the equation [tex]\( 3P = P \cdot e^{rt} \)[/tex].
2. Divide both sides by [tex]\( P \)[/tex]: [tex]\( 3 = e^{rt} \)[/tex].
3. Take the natural logarithm of both sides: [tex]\( \ln(3) = rt \)[/tex].
4. Solve for [tex]\( t \)[/tex]: [tex]\( t = \frac{\ln(3)}{r} \)[/tex].
Given [tex]\( r = 2.5\% = 0.025 \)[/tex]:
[tex]\[ t = \frac{\ln(3)}{0.025} \][/tex]
Using the values:
[tex]\[ \ln(3) \approx 1.099 \][/tex]
[tex]\[ t \approx \frac{1.099}{0.025} \][/tex]
[tex]\[ t \approx 43.94 \][/tex]
Hence, the time required for the amount to triple is approximately 43.94 years.
To summarize:
(a) The time required for the amount to double at an interest rate of 2.5% compounded continuously is approximately 27.73 years.
(b) The time required for the amount to triple at an interest rate of 2.5% compounded continuously is approximately 43.94 years.
