An investment pays 8% interest, compounded annually.
a) write an equation that expresses the amount, A, of the investment as a function of time, t, in years.
b) determine how long it will take for this investment to double in value and then to triple in value.
c) determine the percent increase in value of the account after 5 years and then after 10 years.
d) explain why the answers to parts b and c do not depend on the amount of the initial principal.
Can someone explain to me how to get started? I do not know where to start except for the fact that I need to make an exponential function.
" =1.47
therefore 1.47 % increase??? "
A factor of 1.47 is a 47% increase
100 * final/original = 147/1 =147 %
which is 47 % over the original 100%
after one year
A = Ao (1.08)
after two years
A = Ao (1.08)(1.08)
after three years
A = Ao (1.08)(1.08)(1.08)
after t years
A = Ao (1.08)^t
When is A/Ao = 2?
2 = 1.08^t
log 2 = t log 1.08
.301 = t * .0334
t = 9.01 years to double
That should get you started.
did i do part c right?
c) determine the percent increase in value of the account after:
5 years,
A=(1.08)^5
=1.47
therefore 1.47 % increase???
c) continued:
after 10 years
A=(1.08)^10
=2.16
therefore 2.16% increase?
Am I doing this right?
d) Explain why the answers to parts b and c do not depend on the amount of the initial principal.
2.16 * 100 = 216 % of original
216 % - original 100% = 116% increase
a) The equation that expresses the amount, A, of the investment as a function of time, t, in years is:
A = P * (1 + r)^t
Where:
A = the amount of the investment after t years
P = the initial principal or the amount of the investment at t = 0
r = the annual interest rate expressed as a decimal (0.08 in this case)
t = time in years
b) To determine how long it will take for the investment to double in value, we can set up the following equation:
2P = P * (1 + 0.08)^t
Simplifying the equation, we get:
2 = 1.08^t
Using logarithms, we can solve for t:
t = log(2) / log(1.08)
Similarly, to determine how long it will take for the investment to triple in value, we set up the equation:
3P = P * (1 + 0.08)^t
Simplifying, we get:
3 = 1.08^t
Using logarithms, we can solve for t:
t = log(3) / log(1.08)
c) To determine the percent increase in value of the account after 5 years and 10 years, we can use the formula:
Percent increase = (A - P) / P * 100
For 5 years:
A = P * (1 + 0.08)^5
Percent increase = (A - P) / P * 100
For 10 years:
A = P * (1 + 0.08)^10
Percent increase = (A - P) / P * 100
d) The answers to parts b and c do not depend on the amount of the initial principal because the interest rate and the time it takes to double or triple the value remain constant. The rate of interest and the time required are the primary factors that determine the growth of the investment, regardless of the initial principal.
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.