# 1.3 Estimation

## Overview

TCCNS Course | MATH 1332: Contemporary Mathematics |

UT Austin Course | M 302: Introduction to Mathematics |

# Introduction

**Suggested Resources and Preparation**

**Materials and Technology**

For the instructor: either a computer/projector or a blackboard/whiteboard.

Optional for the instructor: manipulatives in the form of homogenous objects (like marbles or beans) and containers to represent the scales (pairs of cups or rings made of pipe cleaners).

For the student: blank paper or a word processor is sufficient.

**Prerequisite Assumptions**

Students should be able to work with units, either converting units that measure the same quantity or using units to model situations and solve problems. Review source: Khan Academy.

**Lesson Length**

### 50 minutes

**Overview**

Students are briefly introduced to the concept of estimation when solving quantitative problems. One such problem is explained in detail, with related problems and puzzles following.

**Lesson Objectives**

You will understand that:

- Most quantitative problems in the real world require making choices, and generally those choices must be justified. Many times these choices are simplifying assumptions.
- We choose to estimate quantities in order to save time, but also we might choose to estimate a quantity when it is impossible to measure it.

You will be able to:

- Make reasonable simplifying assumptions when solving quantitative problems.
- Justify your simplifying assumptions.
- Solve quantitative problems that require multiplication, division, and/or dimensional analysis to solve.

# Estimation

How many tennis balls will it take to fill up a standard two-car garage?

We could measure this. We could order thousands of tennis balls from Amazon and have them delivered and shove them through the cracked-open door and then count them all.

But that's even sillier than the original question. So instead we're going to *estimate.*

Some estimates are better than others. So it becomes a game of deciding how accurate you want to be, gathering the right data, doing some math, and then convincing other people that you made smart choices and measurements.

# What makes this problem hard to do?

- There's not really a standard size for two-car garages.
- Even if we choose one garage, most likely it is not a perfect rectangular solid space.
- We can't afford enough tennis balls.
- Tennis balls are spherical not cubical. That makes air gaps that we have to count.
- Tennis balls settle. They don't line up like little cubes.
- Tennis balls are soft and might squish under pressure.

# What information do we need?

We need the **volume of the garage** and we need the **volume taken up by a tennis ball** - not the volume of the actual ball. If we had these two numbers, we could use this formula:

\(\frac{\mbox{Volume of garage in cubic feet}}{\mbox{Number of cubic feet needed per ball}}=\mbox{Number of balls}\)

# How accurate do we want to be?

This question is hard to answer. On the one hand, if this were a homework question and you literally bought thousands of tennis balls and went to your instructor's house, they would probably say you were going overboard. On the other hand, if you said "Probably a million or something idk" then your instructor would probably think you were not very convincing. The real answer is somewhere in between and depends on how much time you have, how many resources, and how important the question is.

For example, if we wanted to estimate how much tax to withhold from our paycheck, the IRS only charges a penalty if you are off by more than $1,000. So in this case we know exactly how close we need to be, and as long as we are in that +/- $1,000 window we're okay. If we wanted to estimate the number of COVID-19 cases in Texas hospitals next month, we only know that we want to be as close as possible and we would work as hard as we could to get a close estimate and prove to other people that we were close.

In this case, homework questions are worth points, so a little important, but we aren't really wanting to put in too much work stacking tennis balls or buying equipment, so we're probably going to be on the lazy side.

# How can we make this easier?

We make estimations easier and sometimes possible by making *simplifying assumptions*.

# Terms

- An assumption is something you pretend is true even though you don’t know if it is true or you know it is not true.
- A simplifying assumption is an assumption that make the calculation process go faster without compromising the quality of our solution.

I'm going to make some simplifying assumptions for this problem:

- My garage looks pretty much like a rectangular solid. It probably isn't perfect, and there's a water heater, and there's the garage door opener, and the garage door is not totally flat. But it isn't worth it to me to take the time for those things.
**I will assume that the garage is a rectangular solid.**I am pretending that is true even though it is not. It lets me use the formula for volume of a rectangular solid, so I only have to measure three things. - At the time of recording, I only had one tennis ball at my house. This was tough, but I was unable to find any tennis balls nearby.
**I will assume that the balls act like cubes and do not settle.**This is super false, but it's the best I have right now.

Now that I have a plan, I'm going to gather my information and crunch the numbers:

Video: gathering data and computing an estimate

Volume of garage \(=24.17\times8.17\times20\:=\:3949\:\mbox{ft}^3\)

Volume for each tennis ball \(=.229\times.229\times.229=.012\:\frac{\mbox{ft}^3}{\mbox{ball}}\)

Total number of balls \(=\frac{3949\:\mbox{ft}^3}{.012\:\frac{\mbox{ft}^3}{\mbox{ball}}}=329,083\:\mbox{balls}\)

# Quiz Questions

Question | Answer |

1 | 1,2 |

2 | 3,4 |

3 | 2 |

# Question 1

You own a gas-powered car. You are thinking about buying an electric vehicle. You are estimating how much money you would save per year if you traded in your car for an EV. Which of the following is a simplifying assumption for this calculation? Select all that apply.

- Assume that the number of miles you drive per month is constant at 1000.
- Assume that the price of gas will remain constant at $4 per gallon.
- Assume that the formula for amount of money you spend on gas is multiplying the amount of gas you use by the price per gallon.
- Assume that you will save $2000 per year.

# Question 2

You own a landscaping company and you are providing an estimate to a customer. You need to know how much your costs will be. Which of the following is a simplifying assumption for this calculation? Select all that apply.

- Assume that the lawn is two acres.
- Assume that your costs will be $50.
- Assume that you can mow about three acres per hour in your riding mower.
- Assume that the maintenance on your equipment will remain steady at about $200 per month.
- Assume that the price of gas will remain constant.

# Question 3

You are taking a road trip in a car and you need to estimate how many miles per day you will cover. You know that you drive an average speed of 60 miles per hour (including breaks), and that you will be spending up to nine hours per day driving. Which formula is best for estimating how many miles per day you will be driving?

- \(\frac{60}{9}\) miles per day
- \(60\cdot9\) miles per day
- \(60^9\) miles per day
- \(\frac{60+9}{9}\) miles per day

# Homework Questions

# Question 1.3.1

Each person takes up 14 cubic feet of volume, so the mile-long box will contain up to \(\frac{5280^3}{14}\approx10.5\) billion people. There is plenty of space.

# Question 1.3.2

Simplifying assumptions will vary. Most commonly students choose to have this person working 40 hours per week for 52 weeks or 24/7/265. If the billionaire is working 24/7/365, then they are making \(\frac{1,000,000,000,000}{365\cdot24\cdot60\cdot60}=\$31,710\) per second.

# Question 1.3.3

- It would take \(23,000,000,000,000/10,000,000=2,300,000\) hours. This is equal to \(\frac{2,300,000}{24\cdot365}\approx263\) years. Answers may vary depending on how the students choose to calculate the number of years (leap days, converting to weeks first, etc.) or the units the students choose.
- Likely years are the most impactful units of time for an ordinary person. Or even decades or centuries.
- The national debt is
*about*$23 trillion, and we used that exact number. Also, the legislator is claiming that the cost savings is $10 million per hour, but that is almost certainly an estimate that contains assumptions within it. - This is not really a great plan. A lot of things will change in 263 years, and new costs will almost certainly be incurred in the meantime. Most politicians are looking to have an impact within 2-4 years.

# Question 1.3.4

Answers will vary. Ideally, students are discussing numerical tasks or problems, they are discussing instances of estimation rather than simple guessing (meaning there is some kind of calculation or measurement happening).

# Question 1.3.5

Anything that reasonably fits the description of a quantity that is deliberately simplified in order to ease computation should count as a valid response. Most commonly students will list an assumption which is not numerical or does not ease computation. For example, "I had to do a group project and I assumed that my teammates would contribute equally and they did so we got a high grade."

# Question 1.3.1

Show that you could put the entire world population of humans into a cube that measures 1 mile x 1 mile x 1 mile. You can make the simplifying assumption that all people can fit into a box that is 7' tall, 2' wide and 1' deep.

# Question 1.3.2

There are approximately 2,100 billionaires in the world, and some experts estimate that there may be a trillionaire within our lifetimes. Let's suppose there is a person who makes $1 trillion over the next 30 years. If we pretended this person had an hourly job that paid this, how much money per second would this person be making? List your simplifying assumptions along with your calculations.

# Question 1.3.3

The United States national debt is now about $23 trillion. Suppose a legislator presented a bill to pay down the current national debt, claiming that this cost-saving measure would reduce the debt by $10 million per hour.

- How long would it take to pay off the national debt (ignore new debt that we incur while we are paying this old debt)?
- What units did you choose for your answer in part 1? What units do you think would be the most impactful if you were, say, a journalist?
- What simplifying assumptions were made in the problem?
- In your opinion, is this plan a good plan? Why or why not?

# Question 1.3.4

Think of *two separate times* in your life that you have had to estimate something. Describe the quantity you needed to estimate and write a couple sentences about how accurate you needed the estimate to be.

# Question 1.3.5

Choose one of those estimates and describe at least one simplifying assumption that you made. If you can't think of a simplifying assumption that you actually made, you can describe a simplifying assumption that you could have made.