Question 5 - Add and Subtract Polynomials COLLEGE LIVING The total number of students \( T \) who attend a college consists of two groups: students who live in dorm rooms on campus \( D \) and students who live in apartments off campus \( A \). The number (in hundreds) of students who live in dorm rooms and the total number of students enrolled in the college can be modeled by the following equations, where \( n \) is the number of years since 2001. \[ \begin{array}{l} T=17 n+23 \\ D=11 n+8 \end{array} \] Part A Write an equation that models the number of students who live in apartments. Part B Predict the number of students who will live in apartments in 2020. Part C What do you need to assume in order to predict the number of students who will live on campus in 2020? A. The total number of students does not include students who commute. B. Students do not share dorm rooms. C. The number of students enrolled in the college remains the same. D. Many students live at home during the summer. Question 6 - Solve Equations with Polynomial Expressions Solve each equation. a. \( 2 p=3(4 p-10) \) b. \( -3 q(2 q+5)=2\left(-3 q^{2}+15\right)-5(10 q+6) \)
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To find the number of students living in apartments, we can use the given equations for \( T \) and \( D \): \( A = T - D \). Plugging in the values, we find that \( A = (17n + 23) - (11n + 8) = 6n + 15 \). This means the number of students living in apartments can be modeled by the equation \( A = 6n + 15 \). To predict the number of students living in apartments in 2020, we need to find the value of \( n \) for that year. Since \( n \) represents the number of years since 2001, for 2020, \( n = 19 \). Plugging that into the equation \( A = 6(19) + 15 \) gives us \( A = 114 + 15 = 129 \). Therefore, by 2020, there will be approximately 1,290 students living in apartments! To predict the number of students living on campus in 2020, we need to ensure that the total enrollment does not fluctuate dramatically from existing trends. The most sensible assumption is C: The number of students enrolled in the college remains the same. This ensures our predictions align with established growth patterns observed in the previous years. --- The trend of college students living in various accommodations began to shift significantly around the 1960s. As campuses expanded, more students began opting for the experience of campus life, leading to an increase in on-campus housing and, subsequently, a rise in dorm population. Understanding this historical perspective helps us anticipate future trends in student accommodations. Calculating with polynomials often brings up the age-old challenge of mismanagement in distribution—common mistakes include forgetting to distribute correctly or combining like terms prematurely. A nifty tip is to always double-check your signs and terms before wrapping it up, ensuring you don't miss a crucial negative sign! A systematic approach can keep your calculations as smooth as butter!
