If 50100 dollars is deposited in an account for 14 years at 5.35 percent compounded continuously , find the average value of the account during the 14 years period.
Use
amount = principal * ert
where
principal = initial investment
e=napier's constant=2.7182818284...
r=annual rate of interest
t=time in number of years
Example:
1000$ invested for 3 years at 6%
amount = 1000 * e0.06*3
= $1197.22
To find the average value of the account during the 14-year period, we need to use the formula for continuously compounded interest:
A = P * e^(rt)
Where:
A = Final amount in the account
P = Principal amount (initial deposit)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (in decimal form)
t = Number of years
In this case, the initial deposit (P) is $50,100, the interest rate (r) is 5.35% (or 0.0535 in decimal form), and the number of years (t) is 14.
Let's calculate the average value of the account step-by-step:
Step 1: Convert the interest rate to decimal form:
r = 0.0535
Step 2: Calculate the exponential term:
e^(rt) = e^(0.0535 * 14)
Step 3: Calculate the average value of the account:
A = P * e^(rt)
A = $50,100 * e^(0.0535 * 14)
Now, we can calculate the value of A using a scientific calculator or an online calculator.
To find the average value of the account during the 14-year period, we need to calculate the continuously compounded interest over the entire term.
The formula for continuously compounded interest is given by:
A = P * e^(rt)
Where:
A = the future value of the account
P = the principal amount (initial deposit)
e = the mathematical constant e (approximately 2.71828)
r = the annual interest rate (expressed as a decimal)
t = the time in years
In this case, the initial deposit (principal) is $50,100, the interest rate is 5.35% (0.0535 as a decimal), and the time is 14 years.
Substituting these values into the formula:
A = 50100 * e^(0.0535 * 14)
Now, we can calculate the future value of the account:
A = 50100 * e^(0.749)
Using a scientific calculator or an online calculator that supports the exponential function, we can evaluate this expression to find the future value.
Once we have the future value, to find the average value of the account over the 14-year period, we divide the future value by the total number of years:
Average Value = Future Value / Number of Years
So, in this case, the average value of the account would be:
Average Value = A / 14
Evaluate the expression A, then divide the result by 14 to find the average value of the account during the 14-year period.