What is the future, rounded to whole dollars, of $25,000 after 7 1/2 years, if money earns at an annual rate of 5.75% compounded continuosly?
To calculate the future value of an investment compounded continuously, we can use the continuous compounding formula:
FV = P * e^(r * t)
Where:
FV is the future value
P is the principal amount (initial investment)
e is the base of the natural logarithm (approximately 2.71828)
r is the annual interest rate (as a decimal, so 5.75% = 0.0575)
t is the time period in years
In this case, the principal amount (P) is $25,000, the interest rate (r) is 5.75%, and the time period (t) is 7 1/2 years.
First, we need to convert the mixed number 7 1/2 to a decimal. To do this, we add the whole number part (7) to the fractional part (1/2) which equals 7.5 years.
Now we can substitute the values into the formula:
FV = $25,000 * e^(0.0575 * 7.5)
To calculate this, we need to know the approximate value of e^(0.0575 * 7.5). To find this, we can use a scientific calculator or an online calculator that supports exponentials.
After calculating e^(0.0575 * 7.5), the result will be a number slightly greater than 1. Let's say it's approximately 1.437.
Now we can calculate the future value:
FV = $25,000 * 1.437
Multiplying $25,000 by 1.437 gives us:
FV ≈ $35,925
So, the future value of $25,000 after 7 1/2 years, with an annual interest rate of 5.75% compounded continuously, is approximately $35,925.