how long will it take for an investment to triple if it is compounded continuously at 4.2%
3p = p e^(.042 t)
3 = e^(.042 t)
ln(3) = .042 t
the time units for t are the same as the interest ... yearly, monthly, etc.
To determine how long it will take for an investment to triple with continuous compounding at a rate of 4.2%, you can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = final amount
P = initial investment
r = interest rate (as a decimal)
t = time (in years)
e = Euler's number (approximately 2.71828)
In this case, since we want the investment to triple, the final amount A will be equal to 3 times the initial investment P.
3P = P * e^(0.042t)
Dividing both sides of the equation by P, we get:
3 = e^(0.042t)
Next, take the natural logarithm (ln) of both sides of the equation:
ln(3) = 0.042t * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(3) = 0.042t
Now, solve for t by dividing both sides of the equation by 0.042:
t = ln(3) / 0.042
Using a calculator, we find:
t ≈ 16.47 years
Therefore, it will take approximately 16.47 years for the investment to triple at a continuous compounding rate of 4.2%.
To find out how long it will take for an investment to triple when compounded continuously at a rate of 4.2%, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Final amount (in this case, triple the initial amount)
P = Principal amount (initial investment)
e = Euler's number (~2.71828)
r = Interest rate (in decimal form)
t = Time (in years)
In this case, we want A to be three times the initial investment (P), i.e., A = 3P. We can substitute these values into the formula to solve for t:
3P = P * e^(0.042t)
Divide both sides of the equation by P:
3 = e^(0.042t)
e^(0.042t) = 3
To solve for t, we need to take the natural logarithm (ln) of both sides of the equation:
ln(e^(0.042t)) = ln(3)
0.042t * ln(e) = ln(3)
Since ln(e) is equal to 1, the equation simplifies to:
0.042t = ln(3)
Finally, divide both sides of the equation by 0.042 to solve for t:
t = ln(3) / 0.042
Using a calculator, we can find:
t ≈ 16.48 years
Therefore, it will take approximately 16.48 years for the investment to triple when compounded continuously at a rate of 4.2%.