Objective 3 - Construct Linear Model

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Construct a model equation for the real-life situation.

Scenarios we normally use linear functions to model are:

Cost to produce, Utility Bill, Depreciation of Value;
Relative distance of two objects, Using round-trip times to calculate distance,
Mixing two different concentrations of solutions;
Line of best fit.

After you complete each of the questions below, see if you can put together a “general form” for the linear model you built. You can print out these notes to follow along with the video below and keep notes to organize your thoughts.

A company sells doughnuts. They incur a fixed cost of $ $\sage {fixedCost1}$ for rent, insurance, and other expenses. It costs $ $\sage {productionCost1}$ to produce each doughnut. The company sells each doughnut for $ $\sage {sellingPoint1}$ .

Part A. Construct a linear model that describes their total costs, $C$ , as a function of the number of doughnuts, $x$ , they produce.

$C(x) = \answer {\sage {costs1}}$

Part B. Construct a linear model that describes their total profits, $P$ , as a function of the number of doughnuts, $x$ , they produce.

$P(x) = \answer {\sage {profits1}}$

Part C. Construct a linear model that describes their total revenue, $R$ , as a function of the number of doughnuts, $x$ , they produce.

$R(x) = \answer {\sage {revenue1}}$

Aubrey is a college student going into her first year at UF. She will receive Bright Futures, which covers her tuition plus a $300 educational expense each Fall and Spring semester. Before college, Aubrey saved up $\$\sage {savings2}$ . She knows she will need to pay $\$\sage {rent2}$ in rent a month, $\$\sage {food2}$ for food a week, and $\$\sage {misc2}$ in other weekly expenses.

Part A. Construct a linear model that describes her total costs, $C$ as a function of the number of months, $x$ , she is at UF during Fall semester.

$C(x) = \answer {\sage {costs2}}$

Part B. Construct a linear model that describes her total income, $I$ , as a function of the number of months, $x$ , she is at UF during Fall semester.

$I(x) = \answer {\sage {profits2}}$

Part C. Construct a linear model that describes their total budget, $B$ , as a function of the number of months, $x$ , she is at UF during Fall semester.

$B(x) = \answer {\sage {revenue2}}$

Try to write down notes on how to solve the first two questions in general.

Two UFPD are patrolling the campus on foot. To cover more ground, they split up and begin walking in different directions. Office A is walking at $\sage {officerAspeed3}$ mph while Office B is walking at $\sage {officerBspeed3}$ mph.

Part A. Construct a linear model that describes Officer A’s distance from their starting point, $D$ , as a function of minutes, $m$ , that have passed.

$D(m) = \answer {\sage {partA3}}$

$\text {Speed} = \frac {\text {distance}}{\text {time}}$

Can you re-solve this for distance?

Part B. Construct a linear model that describes their total distance from each other, $T_1$ , as a function of minutes, $m$ , that have passed if they were walking in exactly opposite directions (e.g., North/South).

$T_1(m) = \answer {\sage {partB3}}$

Part C. Construct a linear model that describes their total distance from each other, $T_2$ , as a function of minutes, $m$ , that have passed if they were walking in exactly 90 degrees from each other (e.g., North/East).

$T_2(m) = \answer {\sage {partC3}}$

Exact value needed for the coefficient! DO NOT ROUND.

For Part C, draw a picture and think about how the Pythagorean Theorem could be used. Remember: these are all linear models! So if your model has a non-linear variable, is there a reason we would ignore part of the domain?

Figure 1: Training path.

A bicyclist is training for a race on a hilly path. Their bike keeps track of their speed at any time, but not the distance traveled. Their speed traveling up a hill is $\sage {speedUp4}$ mph, $\sage {speedDown4}$ mph when traveling down a hill, and $\sage {speedFlat4}$ mph when traveling along a flat portion.

Distance is equal to rate times time.

Part A. Construct linear models that describe their distance, $D$ in miles, on a particular portion of the path in terms of the time, $t$ in hours, spent on that part of the path.

$D_{\text {up}}(t) = \answer {\sage {speedUp4*t}}$

$D_{\text {down}}(t) = \answer {\sage {speedDown4*t}}$

$D_{\text {flat}}(t) = \answer {\sage {speedFlat4*t}}$

Part B. Construct linear models that describe their time, $t$ in hours, on a particular portion of the path in terms of the length, $D$ in miles, of that part of the path.

$t_{\text {up}}(D) = \answer {\sage {D/speedUp4}}$

$t_{\text {down}}(D) = \answer {\sage {D/speedDown4}}$

$t_{\text {flat}}(D) = \answer {\sage {D/speedFlat4}}$

Part C. Construct a linear model that describes the total distance of the path, $D$ , in terms of the time spent on a particular path if we knew that the time spent on each path was equal.

$D(t) = \answer {\sage {partC4}}$

Part D. Construct a linear model that describes the total time $T$ spent on the path in terms of the distance of a particular part of the path if we knew that all parts of the path are equal length.

$T(D) = \answer {\sage {partD4}}$

Kappa Delta is hosting an all-you-can-eat pancake fundraiser to support the prevention of child abuse. Adult (18+) tickets are $\$ \sage {adultTicketPrice}$ and teen (10-17) tickets are $\$\sage {teenTicketPrice}$ . Children under 10 are let in without a ticket. The ticket-sellers only kept track of the total number of tickets sold, $\sage {totalTickets}$ , and total revenue, $\$ \sage {totalRevenue}$ .

Part A. Construct a linear model that describes the total number of adult tickets, $y$ , sold in terms of the number of teen tickets, $x$ , sold.

$y = \answer {\sage {eqPartA5}}$

Part B. Construct a linear model that describes the revenue made from selling many adult tickets, $R_a$ , in terms of the number of teen tickets, $x$ , sold.

$R_a = \answer {\sage {eqPartB5}}$

Part C. Construct a linear model that describes the total revenue made, $R_T$ , in terms of the number of teen tickets, $x$ , sold.

$R_T = \answer {\sage {eqPartC5}}$

Part A: Is there way to build an equation that relates adult tickets, teen tickets, and total tickets? Solving this equation for adult tickets would give you the linear model.

Part B: It may be easier to first build this model in terms of $y$ , then use your answer from part A.

Part C: Think about how to model total revenue, then use your answer in Part B to make part of the model.

Chemists commonly create a solution by mixing two products of differing concentrations together. For example, a chemist could have large amounts of a $\sage {concA}\%$ acid solution and a $\sage {concB}\%$ acid solution, but need a $\sage {totalVolume}$ liter $\sage {concTotal}$ % solution.

Part A. Construct a linear model that describes the volume of the $\sage {concB}\%$ acid solution, $v_{\sage {concB}}$ , in terms of the volume of the $\sage {concA}\%$ acid solution, $v$ .

$v_{\sage {concB}} = \answer {\sage {eqPartA6}}$

Part B. Construct a linear model that describes the amount of acid in a $\sage {concA}\%$ acid solution, $A_{\sage {concA}}$ , in terms of the volume of the $\sage {concA}\%$ acid solution, $v$ .

$A_{\sage {concA}} = \answer {\sage {eqPartB6}}$

Part C. Construct a linear model that describes the amount of acid in a $\sage {concB}\%$ acid solution, $A_{\sage {concB}}$ , in terms of the volume of the $\sage {concA}\%$ acid solution, $v$ .

$A_{\sage {concB}} = \answer {\sage {eqPartC6}}$

Part D. Construct a linear model that describes the amount of acid in a $\sage {concTotal}\%$ acid solution, $A_{\sage {concTotal}}$ , in terms of the volume of the $\sage {concA}\%$ acid solution, $v$ .

$A_{\sage {concTotal}} = \answer {\sage {eqPartD6}}$

Parts A-D: Think about what you did in the last problem. How can we use that same structure in this new setting?