To answer these questions, you need to understand how compound interest works and how to derive the exponential function that represents the investment over time.
a) The equation that expresses the amount, A, of the investment as a function of time, t, in years can be derived using the formula for compound interest:
A = P(1 + r)^t
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the interest rate per period (expressed as a decimal)
t = the number of periods (in this case, years)
Since the interest is compounded annually, the interest rate per period is 8% or 0.08.
Therefore, the equation would be: A = P(1 + 0.08)^t
b) To determine how long it will take for the investment to double in value, you need to find the value of t when A is twice the initial investment (2P).
So, you can set up the equation as follows: 2P = P(1 + 0.08)^t
To solve this equation, divide both sides by P:
2 = (1.08)^t
To isolate t, take the logarithm (base 1.08) of both sides:
t = log base 1.08(2)
Similarly, to determine how long it will take for the investment to triple in value, set up the equation as follows: 3P = P(1 + 0.08)^t
Again, divide both sides by P:
3 = (1.08)^t
Take the logarithm (base 1.08) of both sides:
t = log base 1.08(3)
c) To determine the percent increase in value of the account after 5 years, subtract the initial investment (P) from the future value after 5 years (A) and divide by the initial investment, then multiply by 100 to express it as a percentage:
Percent increase = ((A - P) / P) * 100
After 5 years, the equation can be written as:
Percent increase = ((P(1 + 0.08)^5 - P) / P) * 100
Similarly, to determine the percent increase after 10 years, substitute t = 10 in the equation:
Percent increase = ((P(1 + 0.08)^10 - P) / P) * 100
d) The answers to parts b and c do not depend on the amount of the initial principal because we are considering the proportional growth rate which is consistent regardless of the principal amount. The percent increase and the time it takes for the investment to double or triple are determined solely by the interest rate and compounding periods.
Regardless of the initial principal, the rate of return (8%) and the compounding frequency (annual) remain the same, resulting in the same time to double or triple the investment and the same percent increase over a given time period.