## Thinking Mathematically (6th Edition)

(a) The amount can be computed with the help of the compound interest formula, by using the formula given below, where A, r, and t is given. The calculation for the interest when the interest is compounded once in a year, the amount earned in year 1 is as follows: \begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\5,000{{\left(1+\frac{0.055}{1}\right)}^{1}}\\&=\5,000\left(1.055\right)\\&=\5,275\end{align} The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\5,275-\5,000\\&=\275\end{align} The calculation for the interest when the interest is compounded once in a year, the amount earned in year 5 is as follows: \begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\5,000{{\left(1+\frac{0.055}{1}\right)}^{5}}\\&=\5,000{{\left(1.055\right)}^{5}}\\&=\5,000\times1.30696\end{align} $=\6,534.8\approx\6,535$ The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\6,535-\5,000\\&=\1,535\end{align} The calculation for the interest when the interest is compounded once in a year, the amount earned in year 10 is as follows: \begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\5,000{{\left(1+\frac{0.055}{1}\right)}^{10}}\\&=\5,000{{\left(1.055\right)}^{10}}\\&=\5,000\times1.708144\end{align} $=\8,540.72\approx\8,541$ The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\8,541-\5,000\\&=\3,541\end{align} The calculation for the interest when the interest is compounded once in a year, the amount earned in year 15 is as follows: \begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\5,000{{\left(1+\frac{0.055}{1}\right)}^{20}}\\&=\5,000{{\left(1.055\right)}^{20}}\\&=\5,000\times2.917757\end{align} $=\14,588.79\approx\14,589$ The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\14,589-\5,000\\&=\9,589\end{align} (b) The amount can be computed with the help of the compound interest formula, by using the formula given below, where A, r, and t is given. The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 1 is as follows: \begin{align} & A=P{{e}^{r\times t}} \\ & =\5,000{{\left(2.71828\right)}^{0.055\times1}}\\&=\5,000\times1.056541\\&=\5,282.70\approx\5,283\end{align} The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\5,283-\5,000\\&=\283\end{align} The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 5 is as follows: \begin{align} & A=P{{e}^{r\times t}} \\ & =\5,000{{\left(2.71828\right)}^{0.055\times5}}\\&=\5,000\times{{\left(2.71828\right)}^{0.275}}\\&=\5,000\times1.31653\end{align} $=\6,582.65\approx\6,583$ The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\6,583-\5,000\\&=\1,583\end{align} The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 10 is as follows: \begin{align} & A=P{{e}^{r\times t}} \\ & =\5,000{{\left(2.71828\right)}^{0.055\times10}}\\&=\5,000\times{{\left(2.71828\right)}^{0.55}}\\&=\5,000\times1.733252\end{align} $=\8,666.26\approx\8,666$ The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\8,666-\5,000\\&=\3,666\end{align} The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 20 is as follows: \begin{align} & A=P{{e}^{r\times t}} \\ & =\5,000{{\left(2.71828\right)}^{0.055\times20}}\\&=\5,000\times{{\left(2.71828\right)}^{1.1}}\\&=\5,000\times3.004164\end{align} $=\15,020.82\approx\15,021$ The interest is as follows: \begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\15,021-\5,000\\&=\10,021\end{align}