How long, to the nearest year, will it take an investment to triple if it is continuously compounded at 16% per year?
To find out how long it will take for an investment to triple with continuous compounding, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount (in this case, three times the initial amount),
P = the principal amount (initial investment),
r = the annual interest rate (as a decimal),
t = the number of years, and
e = the base of the natural logarithm (approximately 2.71828).
In this case, we want the final amount A to be three times the principal amount P. Let's call the time it takes to triple the investment "t".
So, our equation becomes:
3P = P * e^(0.16t)
To find t, we need to isolate it on one side of the equation. We can divide both sides of the equation by P:
3 = e^(0.16t)
Now, we can take the natural logarithm of both sides to remove the exponential function:
ln(3) = 0.16t
Finally, we can divide both sides of the equation by 0.16 to solve for t:
t = ln(3) / 0.16
Using a calculator, we can find the natural logarithm of 3 and divide it by 0.16 to get the value of t.
By following the steps and plugging in the numbers into the formula, we will find the answer.