Question

neil, a real estate agent, earned a big commission for selling a house. if he puts the money into an account that earns 10% interest compounded continuously, how long will it take for his money to double? round your answer to the nearest month. years and months submit

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the time in years. We want to find the time it takes for the money to double, so $A = 2P$. The interest rate $r = 0.10$ (since 10% = 0.10).

Substitute $A = 2P$ and $r = 0.10$ into the formula:
$$2P = Pe^{0.10t}$$

Step2: Solve for $t$

First, divide both sides of the equation by $P$ (assuming $P
eq0$, which it is since it's money):
$$2 = e^{0.10t}$$

Then, take the natural logarithm of both sides. Recall that $\ln(e^x)=x$, so:
$$\ln(2) = \ln(e^{0.10t})$$
$$\ln(2) = 0.10t$$

Now, solve for $t$ by dividing both sides by $0.10$:
$$t=\frac{\ln(2)}{0.10}$$

Calculate $\ln(2)\approx0.6931$, so:
$$t=\frac{0.6931}{0.10}= 6.931\text{ years}$$

Step3: Convert the decimal part of the year to months

The decimal part of the year is $0.931$ years. To convert this to months, multiply by 12 (since there are 12 months in a year):
$$0.931\times12\approx11.17\text{ months}$$
Rounding to the nearest month, this is 11 months.

Answer:

6 years and 11 months