$2500 is invested in an account that pays 12% interest, compounded continuously. Find the time required for the amount to triple and round the final answer to the nearest tenth of a year.
you want e^.12x = 3
To find the time required for the amount to triple, we can use the formula for continuous compound interest:
A = P*e^(rt)
Where:
A = the final amount
P = the initial principal ($2500)
e = Euler's number (approximately 2.71828)
r = interest rate (12% or 0.12)
t = time (unknown)
In this case, we want the final amount A to be three times the initial amount:
3P = P*e^(0.12t)
Now we can solve for t:
3 = e^(0.12t)
Taking the natural logarithm (ln) of both sides:
ln(3) = ln(e^(0.12t))
Using the property of logarithms, ln(e^x) = x:
ln(3) = 0.12t
Now, divide both sides by 0.12:
t = ln(3) / 0.12
Using a calculator, we can find that ln(3) ≈ 1.09861:
t ≈ 1.09861 / 0.12 ≈ 9.155
Therefore, it will take approximately 9.155 years for the amount to triple. Rounded to the nearest tenth of a year, the answer is 9.2 years.
To find the time required for the amount to triple, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A is the final amount (in this case, 3 times the initial investment)
P is the principal amount (initial investment)
e is Euler's number (approximately equal to 2.71828)
r is the interest rate (in decimal form)
t is the time (in years)
In this case, we have:
A = 3P (since we want the amount to triple)
P = $2500
r = 12% = 0.12 (in decimal form)
So we need to solve for t. Rearranging the formula, we have:
3P = P * e^(0.12t)
Dividing both sides by P, we get:
3 = e^(0.12t)
To isolate t, take the natural logarithm (ln) of both sides:
ln(3) = ln(e^(0.12t))
Using the property that ln(e^x) = x, we can simplify further:
ln(3) = 0.12t
Now, divide both sides by 0.12:
ln(3) / 0.12 = t
Using a calculator, we find:
t ≈ 6.7912 years
Rounding to the nearest tenth of a year, the time required for the amount to triple is approximately 6.8 years.