Chapter 10 - Section 10.2 - Arithmetic Sequences - Exercise Set - Page 1061: 67

The given models overestimate the total dormitory charge by $2$ .

The total dormitory charge for undergraduate education is given by: $\left[ \begin{align} & \left( 4565+5249 \right)+\left( 4824+5413 \right) \\ & +\left( 5046+5629 \right)+\left( 5241+5837 \right) \\ \end{align} \right]=41,804$. The model for dormitory charges at public colleges is given by: ${{a}_{n}}=225n+4357$. Here, $ n=1$ for 2010, $ n=2$ for 2011, $ n=3$ for 2012, and $ n=4$ for 2013. We know that the total dormitory charge at a public college is calculated by; ${{S}_{n}}=\frac{n}{2}\left( {{a}_{1}}+{{a}_{n}} \right)$ For $ n=4$, ${{S}_{4}}=\frac{4}{2}\left( {{a}_{1}}+{{a}_{4}} \right)$ Now, value of ${{a}_{1}}$ is; $\begin{align} & {{a}_{1}}=225\times 1+4357 \\ & =4582 \end{align}$ And, value of ${{a}_{4}}$ is; $\begin{align} & {{a}_{4}}=225\times 4+4357 \\ & =5257 \end{align}$ So, ${{S}_{4}}$ is; $\begin{align} & {{S}_{4}}=\frac{4}{2}\left( {{a}_{1}}+{{a}_{4}} \right) \\ & =\frac{4}{2}\left( 4582+5257 \right) \\ & =19678 \end{align}$ The model for dormitory charges at private colleges is given by ${{a}_{n}}=225n+4357$. Here, $ n=1$ for 2010, $ n=2$ for 2011, $ n=3$ for 2012, and $ n=4$ for 2013. We know that total dormitory charge at private colleges is calculated by ${{S}_{n}}=\frac{n}{2}\left( {{a}_{1}}+{{a}_{n}} \right)$ For $ n=4$, ${{S}_{4}}=\frac{4}{2}\left( {{a}_{1}}+{{a}_{4}} \right)$ Now, the value of ${{a}_{1}}$ is $\begin{align} & {{a}_{1}}=198\times 1+5037 \\ & =5235 \end{align}$ And, the value of ${{a}_{4}}$ is $\begin{align} & {{a}_{4}}=198\times 4+5037 \\ & =5829 \end{align}$ So, ${{S}_{4}}$ is $\begin{align} & {{S}_{4}}=\frac{4}{2}\left( {{a}_{1}}+{{a}_{4}} \right) \\ & =\frac{4}{2}\left( 5235+5829 \right) \\ & =22128 \end{align}$ So, the total dormitory charge in a public and private college is $19,678+22,128=41,806$. So, from the bar graph, the total dormitory charge is 41,804 and from the given models, the total dormitory charge is 41,806 dollars. Hence, the given model overestimate the total dormitory charge by 2.