Question

if $4000 is invested at 1.75% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b) if astd is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by astd

Answer 1

To find the compounded value of a $4000 investment at 1.75% interest over 5 years, we use the compound interest formula adjusting the compounding frequency. For continuous compounding, the investment growth is described by the differential equation dA/dt = rA with the initial condition A(0) = P.

Step-by-step explanation:

If $4000 is invested at 1.75% interest, we can find the value of the investment at the end of 5 years using the compound interest formula A = P(1 + r/n)^(nt), where:

• A is the future value of the investment/loan, including interest
• P is the principal investment amount ($4000)
• r is the annual interest rate (decimal) (0.0175)
• n is the number of times that interest is compounded per year
• t is the time the money is invested for in years (5 years)

We can calculate the compounded interest for:

1. Annually (n = 1)
2. Semiannually (n = 2)
3. Monthly (n = 12)
4. Weekly (n = 52)
5. Daily (n = 365)
6. Continuously (Using the formula A = Pe^(rt))

For continuous compounding, the differential equation that describes the growth of the investment over time is dA/dt = rA, where dA/dt is the rate of change of the investment amount A with respect to time t, and r is the interest rate. The initial condition is A(0) = P, which represents the initial amount invested.

Answer 2

To find the value of the investment at the end of 5 years with different compounding frequencies, we can use the formula for compound interest. The values of the investment at the end of 5 years are: (i) Annually: $4313.07, (ii) Semiannually: $4316.06, (iii) Monthly: $4317.68, (iv) Weekly: $4317.95, (v) Daily: $4317.98, and (vi) Continuously: $4318.07. For the case of continuous compounding, the differential equation dA/dt = rA, where A is the amount of the investment at time t and r is the interest rate, with the initial condition A(0) = $4000.

Step-by-step explanation:

To find the value of the investment at the end of 5 years with different compounding frequencies, we can use the formula for compound interest:

Final Value = Principal * (1 + (Interest Rate / Compounding Frequency)) ^ (Compounding Frequency * Time)

(i) Annually: $4000 * (1 + 0.0175)^5 = $4313.07

(ii) Semiannually: $4000 * (1 + (0.0175 / 2))^(2 * 5) = $4316.06

(iii) Monthly: $4000 * (1 + (0.0175 / 12))^(12 * 5) = $4317.68

(iv) Weekly: $4000 * (1 + (0.0175 / 52))^(52 * 5) = $4317.95

(v) Daily: $4000 * (1 + (0.0175 / 365))^(365 * 5) = $4317.98

(vi) Continuously: $4000 * e^(0.0175 * 5) = $4318.07

(b) The differential equation for continuous compounding is dA/dt = rA, where A is the amount of the investment at time t and r is the interest rate. The initial condition is A(0) = $4000.