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.

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.

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, $$r = 3.325\%$$