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To find the accumulated value of an investment of [tex]$15,000 over 5 years at an interest rate of 5%, we use the compound interest formulas \( A = P \left(1 + \frac{r}{n}\right)^{nt} \) for periodic compounding and \( A = Pe^{rt} \) for continuous compounding.
Given:
- Principal amount, \( P \) = $[/tex]15,000
- Annual interest rate, [tex]\( r \)[/tex] = 5% or 0.05
- Time, [tex]\( t \)[/tex] = 5 years
We solve the problem for three types of compounding: semiannual, quarterly, and continuous.
### a. Compounded Semiannually
For semiannual compounding:
- Compounding frequency, [tex]\( n \)[/tex] = 2 (since it is compounded twice a year)
The formula to use is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Plugging in the numbers:
[tex]\[ A = 15000 \left(1 + \frac{0.05}{2}\right)^{2 \cdot 5} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.025\right)^{10} \][/tex]
[tex]\[ A = 15000 \left(1.025\right)^{10} \][/tex]
After calculating the expression:
[tex]\[ A \approx 19201.27 \][/tex]
Therefore, the accumulated value if the money is compounded semiannually is:
[tex]\[ \boxed{19201.27} \][/tex]
### b. Compounded Quarterly
For quarterly compounding:
- Compounding frequency, [tex]\( n \)[/tex] = 4 (since it is compounded four times a year)
The formula to use is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Plugging in the numbers:
[tex]\[ A = 15000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 5} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.0125\right)^{20} \][/tex]
[tex]\[ A = 15000 \left(1.0125\right)^{20} \][/tex]
After calculating the expression:
[tex]\[ A \approx 19230.56 \][/tex]
Therefore, the accumulated value if the money is compounded quarterly is:
[tex]\[ \boxed{19230.56} \][/tex]
### c. Compounded Continuously
For continuous compounding:
The formula to use is:
[tex]\[ A = Pe^{rt} \][/tex]
Plugging in the numbers:
[tex]\[ A = 15000 \times e^{0.05 \cdot 5} \][/tex]
Since [tex]\( e \)[/tex] is a mathematical constant approximately equal to 2.71828:
[tex]\[ A = 15000 \times e^{0.25} \][/tex]
[tex]\[ A = 15000 \times 1.284025 \][/tex]
After calculating the expression:
[tex]\[ A \approx 19260.38 \][/tex]
Therefore, the accumulated value if the money is compounded continuously is:
[tex]\[ \boxed{19260.38} \][/tex]
In summary:
- Compounded semiannually: [tex]\( 19201.27 \)[/tex]
- Compounded quarterly: [tex]\( 19230.56 \)[/tex]
- Compounded continuously: [tex]\( 19260.38 \)[/tex]
- Annual interest rate, [tex]\( r \)[/tex] = 5% or 0.05
- Time, [tex]\( t \)[/tex] = 5 years
We solve the problem for three types of compounding: semiannual, quarterly, and continuous.
### a. Compounded Semiannually
For semiannual compounding:
- Compounding frequency, [tex]\( n \)[/tex] = 2 (since it is compounded twice a year)
The formula to use is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Plugging in the numbers:
[tex]\[ A = 15000 \left(1 + \frac{0.05}{2}\right)^{2 \cdot 5} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.025\right)^{10} \][/tex]
[tex]\[ A = 15000 \left(1.025\right)^{10} \][/tex]
After calculating the expression:
[tex]\[ A \approx 19201.27 \][/tex]
Therefore, the accumulated value if the money is compounded semiannually is:
[tex]\[ \boxed{19201.27} \][/tex]
### b. Compounded Quarterly
For quarterly compounding:
- Compounding frequency, [tex]\( n \)[/tex] = 4 (since it is compounded four times a year)
The formula to use is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Plugging in the numbers:
[tex]\[ A = 15000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 5} \][/tex]
[tex]\[ A = 15000 \left(1 + 0.0125\right)^{20} \][/tex]
[tex]\[ A = 15000 \left(1.0125\right)^{20} \][/tex]
After calculating the expression:
[tex]\[ A \approx 19230.56 \][/tex]
Therefore, the accumulated value if the money is compounded quarterly is:
[tex]\[ \boxed{19230.56} \][/tex]
### c. Compounded Continuously
For continuous compounding:
The formula to use is:
[tex]\[ A = Pe^{rt} \][/tex]
Plugging in the numbers:
[tex]\[ A = 15000 \times e^{0.05 \cdot 5} \][/tex]
Since [tex]\( e \)[/tex] is a mathematical constant approximately equal to 2.71828:
[tex]\[ A = 15000 \times e^{0.25} \][/tex]
[tex]\[ A = 15000 \times 1.284025 \][/tex]
After calculating the expression:
[tex]\[ A \approx 19260.38 \][/tex]
Therefore, the accumulated value if the money is compounded continuously is:
[tex]\[ \boxed{19260.38} \][/tex]
In summary:
- Compounded semiannually: [tex]\( 19201.27 \)[/tex]
- Compounded quarterly: [tex]\( 19230.56 \)[/tex]
- Compounded continuously: [tex]\( 19260.38 \)[/tex]
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