Suppose that time invest $10,000 in an account that offers are percent annual interest, compounded quarterly if the investment increases to $12,694.34 in five years, find the annual rate of interest

The annual rate of interest is 4.80%Step-by-step explanation:The formula for compound interest, including principal sum is[tex]A=P(1+\frac{r}{n})^{nt}[/tex] where:A is the future value of the investment/loan, including interestP is the principal investment amountr is the annual interest rate (decimal)n is the number of times that interest is compounded per unit tt is the time the money is invested or borrowed for Suppose that time invest $10,000 in an account that offers are percent annual interest, compounded quarterly if the investment increases to $12,694.34 in five years∵ P = $10,000∵ A = $12,694.34∵ n = 4 ⇒ compounded quarterly∵ t = 5 years- Substitute all these values in the formula above∴ [tex]12,694.34=10,000(1+\frac{r}{4})^{4(5)}[/tex]∴ [tex]12,694.34=10,000(1+\frac{r}{4})^{20}[/tex]- Divide both sides by 10,000∴ [tex]1.269434=(1+\frac{r}{4})^{20}[/tex]- Insert ㏒ to both sides∴ [tex]log(1.269434)=log(1+\frac{4}{n})^{20}[/tex]∴ [tex]log(1.269434)=20log(1+\frac{4}{n})[/tex]- Divide both sides by 20∴ [tex]0.00518=log(1+\frac{4}{n})[/tex]- Remember [tex]log_{a}b=c[/tex] can be written as [tex]a^{c}=b[/tex]∵ The base of the ㏒ is 10∴ [tex]10^{0.00518}=(1+\frac{r}{4})[/tex]∴ [tex]1.011998806=1+\frac{r}{4}[/tex]- Subtract 1 from both sides∴ [tex]0.011998806=\frac{r}{4}[/tex]- Multiply both sides by 4∴ 0.04799522 = r∵ r is the rate in decimal- To find the annual rate of interest R% multiply r by 100%∴ R% = 0.04799522 × 100% = 4.799522%∴ R% ≅ 4.80%The annual rate of interest is 4.80%Learn more:You can learn more about interest in brainly.com/question/12773544#LearnwithBrainly