Using the formula for continuous compounding, A = Pe^(rt), it will approximately take 20 years for Allison's investment of $70,000 at an interest rate of 6.2% compounded continuously to grow to $223,200.
To calculate how long it will take for Allison's investment to grow to a specific amount with continuous compounding interest, we use the formula for continuous compounding, which is A = Pert, where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
e is the base of the natural logarithm (~2.71828).
r is the annual interest rate (in decimal).
t is the time in years.
Here, we want to find t when A is $223,200, P is $70,000, and r is 0.062 (which is 6.2% expressed as a decimal).
The formula will look like this:
$223,200 = $70,000e0.062t
First, we divide both sides by $70,000:
e0.062t = 3.18857143
Next, we take the natural logarithm of both sides to solve for t:
ln(e0.062t) = ln(3.18857143)
0.062t = ln(3.18857143)
t = ln(3.18857143) / 0.062
t ≈ 20.03 years
Therefore, to the nearest year, it will take approximately 20 years for Allison's investment to grow to $223,200 at an interest rate of 6.2% compounded continuously.