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Section 5.3 Linear Models

The world we live in runs on data. While the powerful statistical methods used to analyze this data is beyond the scope of this class, we do have the tools in this chapter to interpret the results those tools might give us.

Example 5.22.

Market researchers are trying to understand the demand for fidget spinners among college students. Their analysis of the data they collected estimates that at a price of \(x\) dollars, the number of fidget spinners that will be bought is given by the function \(d(x)=-13x+572\text{.}\) We want to answer the following questions:

  1. As the price increases, what happens to demand (the number of fidget spinners purchased)?

  2. If they give away the spinners for free, how many should they expect students will take?

We can answer these questions by understanding what the numbers in the formula represent. Let's do them one at a time.

  1. As the price increases, what happens to demand (the number of fidget spinners purchased)? Since this is about how the outputs (number of fidget spinners sold, the y-values) change as we change the inputs (the price, the x-values), we need to look at slope. The slope of this linear function is negative, which means it is decreasing. As we increase the price of the fidget spinners, the number sold will decrease.

  2. If they give away the spinners for free, how many should they expect students will take? Since this is about what the output is (the number of fidget spinners, the y-value) when the input is 0 (the price, the x-value), we need to look at the y-intercept. In this function, the y-intercept is 572. So, if they are given away for free, students will take 572 of them, according to this model.

Checkpoint 5.23.

Scientists are studying how quickly a particular kind of tree grows. Based on their measurement, the height of the tree \(t\) years after producing its first fruit \(h(t)=7.3t+81\) inches.

  1. How much does this tree grow each year, based on this model?

  2. How tall is the tree when it produces its first fruit?

  3. Based on this model, how tall do you expect the tree to be 50 years after it produces its first fruit?

Answer.

  1. 7.3 inches

  2. 81 inches

  3. 446 inches

Solution.
How much does this tree grow each year, based on this model? Since this is about how the output (the height) changes based on the input (time), this is a question about slope. Slope is how much the output changes when you increase the input by 1. In our context, that would be how much the height changes during one year, which is exactly what the question is asking. So, the answer to this question is just 7.3 inches.How tall is the tree when it produces its first fruit? In the information we are given, we are told that t is the number of years after producing its first fruit. So, it produces the first fruit when t=0. Therefore, the answer to this question is the output (the height) when t=0. Since h(0)=7.3(0)+81=81, the tree is 81 inches tall when it produces its first fruit.Based on this model, how tall do you expect the tree to be 50 years after it produces its first fruit? Since the inputs to this function are years after producing the first fruit, and the outputs are height, this question is asking what the output is when the input is 50. In otherword, this question is looking for h(50), which we know how to find by pluggin in: h(50)=7.3(50)+81 = 446.

Example 5.24.

Economists are looking at how the price of different goods and services have changed since 2010. The functions they found from their data models are given in the table below.

Table 5.25.
Product Price
Toothpaste \(0.15x+3.47\)
Frozen Pizza \(0.37x+6.82\)
Watermelon \(-0.03x+2.05\)
Plastic Silverware \(-0.12x+4.33\)
Movie Ticket \(1.37x+9.52\)

We want to answer the following questions:

  1. Which products have gotten more expensive over time, and which have gotten cheaper?

  2. Which product has increased in price most quickly?

  3. Which product was the most expensive in 2010?

For each of these, we will need to look at the slope and the y-intercept, so let's isolate those and add them to our table:

Table 5.26.
Product Price Slope y-intercept
Toothpaste \(0.15x+3.47\) \(0.15\) \(3.47\)
Frozen Pizza \(0.37x+6.82\) \(0.37\) \(6.82\)
Watermelon \(-0.03x+2.05\) \(-0.03\) \(2.05\)
Plastic Silverware \(-0.12x+4.33\) \(-0.12\) \(4.33\)
Movie Ticket \(1.37x+9.52\) \(1.37\) \(9.52\)

Now we are ready to answer each question one at a time.

  1. Which products have gotten more expensive over time, and which have gotten cheaper? This is a question about how the output (price) changes based on the input (time), which is slope. Getting more expensive would mean that the price is increasing over time, so the slope is positive. Getting cheaper means that the price is decreasing over time, so the slope is negative. Therefore, the toothpaste, pizza, and movie tickets have gotten more expensive and the watermelon and plastic silverware have gotten cheaper.

  2. Which product has increased in price most quickly? We already figured out that the slope is how we determine whether the price is increasing or decreasing. Since we want to know which is increasing quickest, we want to look for the highest slope. So, the movie tickets are increasing most quickly.

  3. Which product was the most expensive in 2010? In the question, we were told that the models start in 2010, which means that the year 2010 is when \(x=0\text{.}\) So, we want to know the highest output (price) when \(x=0\text{,}\) which is exactly the y-intercept. Looking at that column of the table, we see that the movie tickets were the most expensive in 2010.

Checkpoint 5.27.

Scientists are studying how five town's populations have changed since 2015. Based on their data, they have created the following equations to model the populations:

Table 5.28.
Town Population
Town A \(13x+472\)
Town B \(98x+123\)
Town C \(-27x+866\)
Town D \(53x+657\)
Town E \(-3x+584\)

Use the table above to answer the following questions:

  1. Which town had the largest population in 2015?

  2. Which towns are shrinking?

  3. Which town is shrinking the fastest?

Answer.

  1. Town C

  2. Town C and E

  3. Town C

Solution.

Answering these questions will be easier if we have the slope and y-intercept of each model, so let's add that to our table:

Table 5.29.
Town Population Slope y-intercept
Town A \(13x+472\) \(13\) \(472\)
Town B \(98+123\) \(98\) \(123\)
Town C \(-27+866\) \(-27\) \(866\)
Town D \(53+657\) \(53\) \(657\)
Town E \(-3+584\) \(-3\) \(584\)

Now we are ready to tackle the questions:

  1. Which town had the largest population in 2015? In the question, we were told that the models start in 2015, which means that the year 2015 is when \(x=0\text{.}\) So, we want to know the highest output (population) when \(x=0\text{,}\) which is exactly the y-intercept. Looking at that column of the table, we see that Town C had the largest population in 2015.

  2. Which towns are shrinking? This is a question about how the output (population) changes based on the input (time), which is slope. A shrinking town means that the population is decreasing over time, so the slope is negative. Therefore, towns C and E are shrinking over time.

  3. Which town is shrinking the fastest? We already figured out that the slope is how we determine whether the population is increasing or decreasing. Since we want to know which is decreasing quickest, we want to look for the most negative slope. So, Town C is shrinking most quickly.