Solve the compound interest formula for the interest rate r using the properties of rational exponents. then use the obtained formula to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $10,000 and was worth $14,373.53 after 11 years.

Question
Solve the compound interest formula for the interest rate r using the properties of rational exponents. then use the obtained formula to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $10,000 and was worth $14,373.53 after 11 years.

Answers (1)

2021-01-11
Here, we have
\(A = $14,373.53 , P = $10,000,t = 11\) and \(n = 2\)
Using the compound interest formula, we get
\(\because 14,373.53 = 10,000 (1 + \frac{r}{2})^{2}(11)\)
\(\Rightarrow 14,373.53 = 10,000 (1 + \frac{r}{2})^{22}\)
\(\Rightarrow \frac{14,373.53}{10000} = (1 + \frac{r}{2})^{22}\)
\(\Rightarrow(1+\frac{r}{2})^{22}=\frac{14,373.53}{10000}\)
\(\Rightarrow (1 + \frac{r}{2})^{22} = 1.437353\)
Taking In on both sides, we get
\(\Rightarrow In(\frac{1+r}{2})^{22} = In(1.437353)\)
\(\Rightarrow 22 In (\frac{2+r}{2}) = In(1.437353)\)
\(\Rightarrow In(2 + r) = \frac{In(1.437353)}{22} + In 2\)
\(\Rightarrow (2 + r) = e\frac{In(1.437353)}{22} + In 2)\)
\(\Rightarrow r = e\frac{In(1.437353)}{22} + In 2)\)
\(\Rightarrow r = 0.03325\)
\(\Rightarrow r = 3.325\%\)
Therefore, \(r = 3.325\%\)
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