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Introduction: The Nature of Science and Physics

5 Approximation

Learning Objectives

  • Make reasonable approximations based on given data.

In physics—and in all areas of science—we often need to make educated guesses or approximations about quantities we can’t measure directly. Whether you’re a field biologist estimating the size of a population, or a physiologist calculating approximate energy use in exercise, being able to reason quantitatively is a powerful tool.

Let’s say you’re in the field and want to know how far a group of tagged animals might travel in a day. You may not know the exact distance, but you can estimate it based on speed and average travel time. Similarly, you might estimate the volume of a cell under a microscope or the amount of water lost through evaporation from a plant.

Why Approximation Matters in Biology and Physics

Approximations (or “guesstimates”) are often useful when:

  • You don’t have all the precise data you need

  • You’re planning an experiment or calculating expected outcomes

  • You want to check whether a detailed calculation is realistic

  • You want to rule out unreasonable values or identify measurement errors

In physics, approximations often rely on formulas where the input values (like mass, distance, or time) are known only within a range. In biology, this could include estimating:

  • The mass of an average cell (e.g., 1 nanogram)

  • The volume of water a tree transpires daily

  • The rate of oxygen consumption in a mouse vs. a human

Coming Up: Try It Yourself

Let’s look at two examples next that show how you can use simple numbers, formulas, and reasoning to make quick and useful approximations.

Example 5.1: Approximate the Height of a Building

Can you approximate the height of one of the buildings on your campus, or in your neighborhood? Let us make an approximation based upon the height of a person. In this example, we will calculate the height of a 39-story building.

Strategy

Think about the average height of an adult male. We can approximate the height of the building by scaling up from the height of a person.

Solution

Based on information in the example, we know there are 39 stories in the building. If we use the fact that the height of one story is approximately equal to about the length of two adult humans (each human is about 2-m tall), then we can estimate the total height of the building to be

[latex]\frac{\text{2 m}}{\text{1 person}}×\frac{\text{2 person}}{\text{1 story}}×\text{39 stories = 156 m}\text{.}[/latex]

Discussion

You can use known quantities to determine an approximate measurement of unknown quantities. If your hand measures 10 cm across, how many hand lengths equal the width of your desk? What other measurements can you approximate besides length?

Example 5.2: Approximating Vast Numbers: a Trillion Dollars

A bank stack containing one hundred hundred-dollar bills.
Figure 5.1: A bank stack contains one-hundred $100 bills, and is worth $10,000. How many bank stacks make up a trillion dollars? (credit: Andrew Magill)

The U.S. federal deficit in the 2008 fiscal year was a little greater than $10 trillion. Most of us do not have any concept of how much even one trillion actually is. Suppose that you were given a trillion dollars in $100 bills. If you made 100-bill stacks and used them to evenly cover a football field (between the end zones), make an approximation of how high the money pile would become. (We will use feet/inches rather than meters here because football fields are measured in yards.) One of your friends says 3 in., while another says 10 ft. What do you think?

Strategy

When you imagine the situation, you probably envision thousands of small stacks of 100 wrapped $100 bills, such as you might see in movies or at a bank. Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of 100 bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height.

Solution

  1. Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is:
    [latex]\begin{array}{lr}\text{volume of stack}=\text{length}×\text{width}×\text{height,}\\ \text{volume of stack}=\text{6 in}\text{.}×\text{3 in}\text{.}×0\text{.}\text{5 in}\text{.},\\ \text{volume of stack}=\text{9 in}{\text{.}}^{3}\text{.}\end{array}[/latex]
  2. Calculate the number of stacks. Note that a trillion dollars is equal to [latex]$1×{\text{10}}^{\text{12}},[/latex] and a stack of one-hundred [latex]$\text{100}[/latex] bills is equal to [latex]$\text{10},\text{000},[/latex] or [latex]$1×{\text{10}}^{4}[/latex]. The number of stacks you will have is:
    [latex]\$1×{\text{10}}^{\text{12}}\left(\text{a trillion dollars}\right)\text{/\$1}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{per stack}=1×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{stacks.}[/latex]
  3. Calculate the area of a football field in square inches. The area of a football field is [latex]\text{100 yd}×\text{50 yd,}[/latex] which gives [latex]5,{\text{000 yd}}^{2}.[/latex] Because we are working in inches, we need to convert square yards to square inches:
    [latex]\begin{array}{}\text{Area}={\text{5,000 yd}}^{2}×\frac{3\phantom{\rule{0.25em}{0ex}}\text{ft}}{\text{1 yd}}×\frac{3\phantom{\rule{0.25em}{0ex}}\text{ft}}{\text{1 yd}}×\frac{\text{12}\phantom{\rule{0.25em}{0ex}}\text{in}\text{.}}{\text{1 ft}}×\frac{\text{12}\phantom{\rule{0.25em}{0ex}}\text{in}\text{.}}{\text{1 ft}}=6,\text{480},\text{000}\phantom{\rule{0.25em}{0ex}}\text{in}{\text{.}}^{2},\\ \text{Area}\approx 6×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{in}{\text{.}}^{2}\end{array}[/latex]
    This conversion gives us [latex]6×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{in}{\text{.}}^{2}[/latex] for the area of the field. (Note that we are using only one significant figure in these calculations.)
  4. Calculate the total volume of the bills. The volume of all the [latex]$\text{100}[/latex]-bill stacks is [latex]9\phantom{\rule{0.25em}{0ex}}\text{in}{\text{.}}^{3}/\text{stack}×{\text{10}}^{8}\text{ stacks}=9×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{in}{\text{.}}^{3}.[/latex]
  5. Calculate the height. To determine the height of the bills, use the equation:
    [latex]\begin{array}{lll}\text{volume of bills}& =& \text{area of field}×\text{height of money:}\\ \text{Height of money}& =& \frac{\text{volume of bills}}{\text{area of field}},\\ \text{Height of money}& =& \frac{9×{\text{10}}^{8}\text{in}{\text{.}}^{3}}{6×{\text{10}}^{6}{\text{in.}}^{2}}=1.33×{\text{10}}^{2}\text{in.,}\\ \text{Height of money}& \approx & 1×{\text{10}}^{2}\text{in.}=\text{100 in.}\end{array}[/latex]
    The height of the money will be about 100 in. high. Converting this value to feet gives
    [latex]\text{100 in}\text{.}×\frac{\text{1 ft}}{\text{12 in}\text{.}}=8\text{.}\text{33 ft}\approx \text{8 ft.}[/latex]

Discussion

The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough “guesstimates” versus carefully calculated approximations?

Check Your Understanding

Question:
Using mental math and your understanding of fundamental units, approximate the area of a regulation basketball court. Describe the process you used to arrive at your final approximation.

Sample Reasoning and Answer:

Let’s start with a reference unit: an average adult male is about 2 meters tall.

Now imagine lining people up:

  • Length of the court: About 15 people laid end to end, which gives approximately

     

    15×2=30meters 

  • Width of the court: About 7 people side by side, giving

     

    7×2=14meters 

Now estimate the area using:

 

Area=length×width=30m×14m=420m2\text{Area} = \text{length} \times \text{width} = 30\ \text{m} \times 14\ \text{m} = 420\ \text{m}^2Final Approximation:

Approximate area of the court≈420m2 

Scientific Thinking Tip:

In biology or ecology, this same approach is used all the time. For example, you might:

  • Estimate the canopy area of a tree based on trunk spacing

  • Approximate coral reef coverage in square meters from aerial images

  • Model the area of a petri dish based on diameter

The key is not perfect precision—it’s developing confidence in making reasonable, scale-appropriate estimates using familiar references and fundamental units (like meters, grams, liters, etc.).

Summary

  • Scientists often approximate the values of quantities to perform calculations and analyze systems.

Glossary

approximation
an estimated value based on prior experience and reasoning
definition

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College Physics 1 Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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