How long will it take an investment to triple in value if the interest rate is 4% compounded continuously?
i know its set up as 3x=x(e^rt)
but im confused as what to do nxt
3x = x*(e^rt) is correct, and in this case r = 0.04. Take the natural logarithms of both sides of the equation.
ln 3 + ln x = ln x + rt
The x terms drop out.
t = ln3/r = 1.0986/0.04 = 27.46 years
To determine how long it will take for an investment to triple in value with continuous compounding, you can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Final amount or future value
P = Initial investment or present value
r = Interest rate per period (expressed as a decimal)
t = Time in years
e = Euler's number, approximately equal to 2.71828
In this case, since we want the investment to triple, the final value (A) is 3 times the initial value (P). So, we can rewrite the formula as:
3P = P * e^(rt)
Next, we need to solve for t. Let's break down the steps:
1. Divide both sides of the equation by P:
3 = e^(rt)
2. Take the natural logarithm (ln) of both sides:
ln(3) = ln(e^(rt))
Remember that ln(e^(rt)) simplifies to rt:
ln(3) = rt
3. Divide both sides by r:
t = ln(3) / r
Finally, substitute the given interest rate of 4% (or 0.04) into the formula:
t = ln(3) / 0.04
Using a calculator, evaluate ln(3) and divide the result by 0.04 to find the value of t. This will give you the time it will take for the investment to triple in value with a 4% continuous interest rate.