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Sagot :
To address the given problem step-by-step:
1. Initial Investment (Principal):
The initial amount invested is [tex]\( \$170 \)[/tex].
2. Annual Interest Rate (APR):
The annual interest rate is [tex]\( 2.7\% \)[/tex].
3. Continuous Compounding Formula:
When interest is compounded continuously, the value of the investment after [tex]\( t \)[/tex] years is given by the formula:
[tex]\[ A(t) = P \cdot e^{(r \cdot t)} \][/tex]
where:
- [tex]\( A(t) \)[/tex] is the amount after [tex]\( t \)[/tex] years,
- [tex]\( P \)[/tex] is the principal (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the number of years,
- [tex]\( e \)[/tex] is the base of the natural logarithm.
For our specific values:
[tex]\[ P = 170, \quad r = \frac{2.7}{100} = 0.027 \][/tex]
4. Function for the Value of the Account after [tex]\( t \)[/tex] Years:
Substituting the given values into the formula, we get:
[tex]\[ f(t) = 170 \cdot e^{(0.027 \cdot t)} \][/tex]
The coefficient needs to be rounded to four decimal places (though in this formula, it does not change):
[tex]\[ f(t) = 170 \cdot e^{(0.0270 \cdot t)} \][/tex]
5. Annual Percentage Yield (APY):
The APY is the actual percentage increase in the account value over one year, considering the effect of compound interest. For continuous compounding, APY is given by:
[tex]\[ APY = \left( e^{r} - 1 \right) \times 100\% \][/tex]
Substituting [tex]\( r = 0.027 \)[/tex]:
[tex]\[ APY = \left( e^{0.027} - 1 \right) \times 100\% \][/tex]
Calculating the value (already known):
[tex]\[ APY \approx 2.74\% \][/tex]
Final Answer:
1. Function showing the value of the account after [tex]\( t \)[/tex] years:
[tex]\[ f(t) = 170 \cdot e^{0.0270 t} \][/tex]
2. Percentage growth per year (APY):
[tex]\[ APY \approx 2.74\% \][/tex]
This is a detailed, step-by-step solution for the given problem.
1. Initial Investment (Principal):
The initial amount invested is [tex]\( \$170 \)[/tex].
2. Annual Interest Rate (APR):
The annual interest rate is [tex]\( 2.7\% \)[/tex].
3. Continuous Compounding Formula:
When interest is compounded continuously, the value of the investment after [tex]\( t \)[/tex] years is given by the formula:
[tex]\[ A(t) = P \cdot e^{(r \cdot t)} \][/tex]
where:
- [tex]\( A(t) \)[/tex] is the amount after [tex]\( t \)[/tex] years,
- [tex]\( P \)[/tex] is the principal (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the number of years,
- [tex]\( e \)[/tex] is the base of the natural logarithm.
For our specific values:
[tex]\[ P = 170, \quad r = \frac{2.7}{100} = 0.027 \][/tex]
4. Function for the Value of the Account after [tex]\( t \)[/tex] Years:
Substituting the given values into the formula, we get:
[tex]\[ f(t) = 170 \cdot e^{(0.027 \cdot t)} \][/tex]
The coefficient needs to be rounded to four decimal places (though in this formula, it does not change):
[tex]\[ f(t) = 170 \cdot e^{(0.0270 \cdot t)} \][/tex]
5. Annual Percentage Yield (APY):
The APY is the actual percentage increase in the account value over one year, considering the effect of compound interest. For continuous compounding, APY is given by:
[tex]\[ APY = \left( e^{r} - 1 \right) \times 100\% \][/tex]
Substituting [tex]\( r = 0.027 \)[/tex]:
[tex]\[ APY = \left( e^{0.027} - 1 \right) \times 100\% \][/tex]
Calculating the value (already known):
[tex]\[ APY \approx 2.74\% \][/tex]
Final Answer:
1. Function showing the value of the account after [tex]\( t \)[/tex] years:
[tex]\[ f(t) = 170 \cdot e^{0.0270 t} \][/tex]
2. Percentage growth per year (APY):
[tex]\[ APY \approx 2.74\% \][/tex]
This is a detailed, step-by-step solution for the given problem.
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