Question

Suppose that $1000 is initially invested in an account at a fixed interest rate, compounded continuously. Suppose also that, after five years, the amount of money in the account is $1207. Find the interest rate per year. Write your answer as a percentage. Do not round any intermediate computations, and round your percentage to the nearest hundredth.

Answer 1

The interest rate per year is 3.76%

Explanation:

Given the following parameters

Initial amount = $1000

time = 5 years

Final amount = $1207

Interest rate =?

To find the interest rate, we will be applying the continuous compounding formula

`\begin{gathered} The\text{ continuous compounding formula is given as} \\ A\text{ = P}e^{r\cdot\text{ t}} \\ \text{Where A = final amount} \\ P\text{ = initial amount} \\ r\text{ = interest rate} \\ t\text{ = time} \\ \text{Substitute the above parameters into the formula} \\ 1207\text{ = 1000 }e^{r\cdot\text{ 5}} \\ \text{Divide both sides by 1000} \\ (1207)/(1000)\text{ }\frac{1000\text{ }e^(5r)}{1000} \\ 1.207\text{ = }e^(5r) \\ To\text{ find r, take the natural log of both sides} \\ \ln \text{ (1.207) = }\ln \text{ (}e^(5r)) \\ \ln \text{ (1.207) = 5r} \\ 0.188138\text{ = 5r} \\ \text{Divide both sides by 5} \\ (0.188138)/(5)\text{ = }(5r)/(5) \\ r\text{ = 0.03763} \\ Convert\text{ the interest rate to \% by multiplying by 100} \\ r\text{ = 0.03763 }\cdot\text{ 100\%} \\ r\text{ = 3.76\%} \\ \text{Hence, the interest rate per year is 3.76\%} \end{gathered}`