Section 1 Linear Functions and Their Modeling
Task 1.1.
Suppose that you work for a landscape designer Nancy. Her specialty is designing square ponds that are surrounded by hand-painted tiles. Customers can order the pond in any size square, starting with sides measuring \(2\) feet, and available in one-foot increments thereafter (sides of \(3\) feet, \(4\) feet, \(5\) feet, etc.). The tiles are \(1\)-foot square, and are placed edge-to-edge along the entire outer perimeter of the pond.- Draw a picture that describes a \(2\) by \(2\) square pond, clearly indicating the portion of the pond and the portion of the tiles. How many tiles are needed to cover the border for this \(2\) by \(2\) square pond?
- Repeat the last question for different pond sizes, from \(3\) by \(3\) up through \(6\) by \(6\text{.}\) Make a table showing the number of border tiles needed for each size.
- Based on the table that you created in the last part, if the side length of the pond is increased by \(1\) foot, how many more border tiles are needed? Explain why the number of tiles is increasing according to this pattern.
- How many border tiles are needed for a pond with sides of length \(12\) feet?
- If Nancy orders \(64\) border tiles for an upcoming job, how large is the pond the customer wants?
- Write an equation for the number of tiles as a function of the length of one edge of the pond.
The function that gives the number of tiles as a function of the length of one edge of the pond in the last problem is called a linear function. A linear function is of the form
\begin{equation*}
y=mx+b
\end{equation*}
where \(m\) is called the slope and \(b\) is the \(y\)-intercept of the function. If \((x_1,y_1), (x_2,y_2)\) are two different points on a linear function with \(x_1 \neq x_2\text{,}\) then the slope of this linear function can be calculated using the formula
\begin{equation*}
m = \frac{y_2-y_1}{x_2-x_1}
\end{equation*}
Task 1.2.
Suppose we know that there are 3718 students at Utica College in 2012 and 4463 students at Utica College in 2016. Assume that the student growth at Utica College follow a linear model.- Write an equation for the number of students as a function of years using the number of students in year 2012 and 2016 (do not use the numbers in the following table for this question).
- Below is a table that gives the number of students at Utica College from year 2012 to 2016. Go to the link https://www.desmos.com/calculator/st7mr7otcx for a scatterplot of the data in the above table. Graph your function from the last question in the same coordinate system. How well is your function predicting the number of students at Utica College between 2012 and 2016?
Table 1.3. Number of students at Utica College between 2012 and 2016 Year 2012 2013 2014 2015 2016 Number of students 3718 3839 4028 4249 4463 - According to this model, how many students are there in year 2017 at Utica College?
- After what year will the number of students at Utica College be at least \(10,000\text{?}\) Do you think that this is a reasonable estimation?
Task 1.4.
In this task, we will figure out the hourly wage necessary for a family in Utica-Rome area to afford housing. We will look at real data about hourly wages and the cost for renting each month. Your goal is to use mathematics to decide whether or not you think five families in Utica-Rome area are paid fair wages. The following table contains median market rent for different sizes of apartments in the Utica-Rome area in 2021.| Apartment Size | Studio | 1 Bedroom | 2 Bedrooms | 3 Bedrooms | 4 Bedrooms |
| Price | \(\$624\) | \(\$664\) | \(\$847\) | \(\$1054\) | \(\$1213\) |
- RED Family
- 1 adult. You are a Vietnamese American female who is a full time student working about 20 hours per week. You have a minimum wage job working in the library (no tips). But you got a scholarship, so you get $1000 at the beginning of every month.
- GREEN Family
- 1 adult; 1 child. You are a young single white mom with one child working as a server at a nearby restaurant. Minimum wage is different if you receive tips - $8.35 per hour. You make minimum way, and you average about $350 per week in tips. You work 40 hours per week.
- BLUE Family
- 2 adults; 2 children. You are a Latino family with two children under the age of 5. You can’t afford to put both children in childcare. Mom stays home to take care of the children. Dad works 40 hours per week at a construction company that pays 2 times minimum wage for non-tipped employees.
- ORANGE Family
- 1 adult. You are a young Black woman who is going to school part time and working full time (40 hours per week). You work at the same construction company as the dad of the BLUE family, but most Black women (including you) make 64% what men at the company make.
- PURPLE Family
- 2 adults; 3 children. You are a family with three children. All three of your children are in school, so both parents works full time (40 hours per week). Both found jobs working for Amazon fullfillment center in Frankfort, NY. Amazon pays employees $17 per hour.
- Each member of the team will select one family to begin with. If your team has fewer than 5 members, come back to the remaining families after you have completed with your initial choice.
- For each family, create an equation that shows the family's earnings over time. Carefully choose your input and output variable. Then graph the equation in Desmos. Use the color of each family for the graph.
- Determine the type of apartment you think is the best for each family as a team. Who needs to work the fewest hours each month to pay rent? How did you determine this answer?
- Who needs to work the most hours each month to pay rent? How did you determine your answer?
- The purple line and the red line intersect at a point. What is the significance of this point?
- Financial advisors recommend that you only use 30% of your montly income to pay for rent. Do the numbers of hours you found needed to pay rent match 30% of the earnings for each family? If not, what kind of hoursing can the family afford with only 30% of their income?
- If we increase minimum wage to $15.00 per hour for non-tipped employees and $10.50 for tipped employees, how would that change the housing situation for each of the five families?