How long will it take you to get $ 50,000 , if you invested $ 5000 in an account giving 8.7 interest , compounded continuously.
Round your answer to the nearest tenth.
5000 e^(.087t) = 50000
solve for t
To determine how long it will take to accumulate $50,000, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A is the accumulated amount (50,000),
P is the principal amount (5,000),
r is the interest rate (8.7% expressed as a decimal, which is 0.087),
and t is the time in years.
Substituting the given values, the formula becomes:
50,000 = 5,000 * e^(0.087t)
Divide both sides by 5,000 to isolate the exponential term:
10 = e^(0.087t)
Next, take the natural logarithm of both sides:
ln(10) = 0.087t
Now, divide both sides by 0.087:
ln(10) / 0.087 = t
Using a calculator, the value of t is approximately 4.2654072.
Therefore, it will take approximately 4.3 years to accumulate $50,000 with continuous compounding.
To calculate the time it will take to reach $50,000 with continuous compounding, we can use the formula:
A = P * e^(rt)
Where:
A is the final amount ($50,000)
P is the principal amount ($5,000)
e is the mathematical constant approximately equal to 2.71828
r is the interest rate (8.7% or 0.087)
t is the time in years (which we need to find)
Rearranging the formula to solve for t, we have:
t = ln(A/P) / r
Now we can substitute the given values into the formula and calculate:
t = ln(50000/5000) / 0.087
Using a calculator, we can find that ln(50000/5000) ≈ 2.9957.
t ≈ 2.9957 / 0.087
Calculating this expression gives t ≈ 34.4 years.
Therefore, it will take approximately 34.4 years to accumulate $50,000 with continuous compounding at an interest rate of 8.7%.