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Mathematics in Trades and Life

Megan Ossiander-Gobeille, Mark Fitch

Section 2.7 Working with Applications

We are emphasizing using math to address questions that arise from various circumstances. This means the problems do not come with a model (think equation or function) and instructions on using it. In this section we will consider how we can go about using a description (typically verbal) to identify and use a model.

Subsection 2.7.1 Process Overview

Our first task is to read the problem to understand it.
  • Read the problem description a few times.
    • If you can paraphrase it, you understand it enough.
    • Drawing a picture and labeling parts may help.
  • Identify what we are asked to do.
  • Identify the information we are given. Note distinctions like measurements and rates.
  • Identify any units. These often help us set up a model.
  • Write everything! We do not model in our heads.
Next we write the mathematical model (equation or function).
  • Use the description to determine which application type (e.g., percent, proportion, linear model, etc.). Note units can suggest this (e.g., meters and meters squared indicate something was squared).
  • Do not insert any numbers yet.
  • Do not do any calculations yet.
Now we will have a model that matches our situation and possibly some numbers to insert.
  • Insert numbers into the model. You may have to calculate some of these (e.g., you are given two points but not the slope you need).
  • Solve for the desired value. Note it may help to do some calculations with the numbers first.
  • State your answer and use units appropriately.
Finally we should check that our answer makes sense. We should not have negative prices (usually) or distances larger than the earth (when working with terrestrial problems).

Example 2.7.1.

You moved across town and rented a 20 foot moving truck for the day. You want to make sure the bill you received is correct. If you paid $81.03, for how many miles were you charged? Assume there were no extra fees.
Solution.
We want to compare the bill we received to the price listed in the add. The question is about how many miles (not how much money).
We are given the price per mile ($0.79 per mile). There is also a fixed cost for the rental ($39.95). Adding the fixed cost and the milage cost will give us the total.
Our model is \(C = \$39.95+\$0.79m\) where \(C\) is the total cost and \(m\) is the number of miles.
We know the total cost, which will leaves \(m\) in the equation, the number of miles, which is what we want to calculate. We can use the solving technique in Section 2.6
\begin{align*} \$81.03 & = \$39.95+\$0.79m\\ -\$39.95+\$81.03 & = -\$39.95+\$39.95+\$0.79m\\ \$41.08 & = \$0.79m\\ \frac{\$41.08}{\$0.79} & = \frac{\$0.79}{\$0.79}m\\ 52 & = m\text{.} \end{align*}
The charge is for 52 miles.
Figure 2.7.2. Using math modeling for rental truck

Exercises 2.7.2 Exercises

1. Minimum Grade.

2. Triangle Properties.

3. Bike Rental.

4. Pilot Training.

5. Thunder.

6. Speeding.

7. Area of Triangle.

8. Unit cost.

9. Paychecks.

10. Mixture.

11. Area of Rectangle.

12. Shelving.

13. Fax Cost.

14. Population Decrease.

15. Sales.

16. Bookcase.

17. Knitting.

18. Rental Cost.

19. Mixture.

20. Mixture.

21. Wire Cutting.

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