1.
You want to invest $10,000 dollars. How long will it take to double your investment at an annual interest rate of 10%, compounded continuously? (Round your answer to the nearest year)
solve for t with
e^(.10*t) = 2
To find out how long it will take to double your investment at an annual interest rate of 10%, compounded continuously, you can use the continuous compound interest formula:
A = P * e^(rt)
where:
A = the future amount or value of the investment
P = the initial principal or investment amount
e = Euler's number (approximately 2.71828)
r = the annual interest rate (as a decimal)
t = the time in years
In this case, you want to find the time it takes for the investment to double, so the future amount (A) will be 2 times the initial investment (P), and you'll use an interest rate (r) of 10% (or 0.10 as a decimal). Rearranging the formula:
2P = P * e^(0.10t)
Dividing both sides by P:
2 = e^(0.10t)
Now, we can take the natural logarithm (ln) of both sides to isolate the exponent:
ln(2) = ln(e^(0.10t))
Using the property of logarithms, ln(e^x) = x:
ln(2) = 0.10t
Finally, we can solve for t by dividing both sides by 0.10:
t = ln(2) / 0.10
Using this formula, you can calculate the value of t:
t ≈ ln(2) / 0.10 ≈ 6.93 years
Rounding to the nearest year, it will take approximately 7 years to double your investment.