Respuesta :
Answer
The interest rate per year is 3.76%
Step-by-step 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
[tex]\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} \\ \frac{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} \\ \frac{0.188138}{5}\text{ = }\frac{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}[/tex]
