Final answer:
To find the value of the investment at the end of 9 years with different compounding periods, use the formula Final Amount = Initial Amount(1 + rac{r}{n})^(n*t). For continuous compounding, use the formula Final Amount = $1000e^{0.04*9}. The initial condition satisfied by A(t) is A(0) = $1000.
Step-by-step explanation:
To find the value of the investment at the end of 9 years with different compounding periods, we can use the formula:
Final Amount = Initial Amount(1 + rac{r}{n})^(n*t)
Where:
Initial Amount = $1000 (given)
r = 4% = 0.04
t = 9 years (given)
n = number of times the interest is compounded per year
(i) Annually:
Using the formula above with n = 1, the final amount is:
Final Amount = $1000(1 + rac{0.04}{1})^(1*9) = $1000(1.04)^9
= $1000 * 1.42240362812
≈ $1422.40
(ii) Semiannually:
Using the formula above with n = 2, the final amount is:
Final Amount = $1000(1 + rac{0.04}{2})^(2*9) = $1000(1.02)^18
≈ $1424.79
(iii) Monthly:
Using the formula above with n = 12, the final amount is:
Final Amount = $1000(1 + rac{0.04}{12})^(12*9) = $1000(1.003333)^108
≈ $1425.23
(iv) Weekly:
Using the formula above with n = 52, the final amount is:
Final Amount = $1000(1 + rac{0.04}{52})^(52*9) = $1000(1.000769)^468
≈ $1425.38
(v) Daily:
Using the formula above with n = 365, the final amount is:
Final Amount = $1000(1 + rac{0.04}{365})^(365*9) = $1000(1.00010958904)^3285
≈ $1425.39
(vi) Continuously:
Using the formula for continuous compounding, the final amount is:
Final Amount = $1000e^{0.04*9} = $1000e^{0.36}
≈ $1425.62
(b) To find the differential equation for continuous compounding, we can use the formula:
rac{dA}{dt} = rA
Where:
A(t) is the amount of the investment at time t
r = 0.04 (given interest rate)
The initial condition satisfied by A(t) is A(0) = $1000 since that is the initial amount of the investment.