Question

(a) If $1000 is invested at 4% interest, find the value of the investment at the end of 9 years if the interest is compounded as follows. (Round your answers to the nearest cent.) (i) annually $ (ii) semiannually $ (iii) monthly $ (iv) weekly $ (v) daily $ (vi) continuously $

(b) If A(t) is the amount of the investment at time t for the case of continuous compounding, write a differential equation satisfied by A(t). dA/dt = Find the initial condition satisfied by A(t). A(0) =

Answer 1

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.