How long will it take a $400 investment to be worth $700 if it is continuously compounded at 12% per year?
I got the answer 6.71 years, but it was wrong, using the A=Pe^rt
Thank you
Pt = Po*e^rt.
r = 12%/100% = 0.12 = APR expressed as a decimal.
t = Time in yrs.
Pt = 400*e^0.12t = 700.
e^0.12t = 700 / 400 = 1.75.
0.12t = 0.55962.(Hit Ln key).
t = 4.66 yrs.
OR
e^0.12t = 700 / 400 = 1.75.
Take Ln of both sides:
0.12t*Ln(e) = Ln1.75.
0.12t*1.0 = 0.55962.
t = 0.55962 / 0.12 = 4.66 yrs.
To find the time it takes for an investment to grow from $400 to $700, continuously compounded at a rate of 12% per year, we need to use the formula for continuous compound growth:
A = P * e^(rt)
Where:
A = the final amount (in this case, $700)
P = the initial amount (in this case, $400)
r = the interest rate (12% or 0.12 in decimal form)
t = the time period we want to find
Substituting these values into the formula, we have:
$700 = $400 * e^(0.12t)
Divide both sides of the equation by $400:
$700 / $400 = e^(0.12t)
Simplifying further:
1.75 = e^(0.12t)
To find the value of 't', we need to take the natural logarithm (ln) of both sides of the equation:
ln(1.75) = ln(e^(0.12t))
Using the property of logarithms that ln(e^(0.12t)) = 0.12t:
0.12t = ln(1.75)
Now divide both sides of the equation by 0.12:
t = ln(1.75) / 0.12
Using a calculator, the approximate value of ln(1.75) is 0.5596. Dividing this by 0.12, we get:
t ≈ 0.5596 / 0.12
Calculating this, we find that t ≈ 4.6633 years.
So, it will take approximately 4.6633 years for a $400 investment at a continuous compound interest rate of 12% per year to grow to $700.