■ How many cubic yards of concrete are needed to pave 1 mile of interstate highway (two lanes each direction)?
■ How many feet of wire are needed to connect the lighting systems in an automobile?
■ What is the average flow rate in gallons per minute of gasoline moving from the fuel tank to the fuel injectors in an automobile cruising at highway speed?
An accomplished engineer knows the answer to most problems before doing any calculations. This does not mean an answer to three significant figures, but a general idea of the range of values that would be reasonable. For example, if you throw a base-ball as high as possible, how long will it take for the base-ball to hit the ground? Obviously, an answer of a few milliseconds is unreasonable, as is several months. Several seconds seems more realistic. When you actually do a calculation to determine this time, you should ask yourself, “Is my answer reasonable?”
Sample Fermi Problems
■ Estimate the total number of hairs on your head.
■ Estimate the number of drops of water in all of the Great Lakes.
■ Estimate the number of piano tuners in New York City.
Every year, numerous people run out of fuel while driving their vehicle on the road.
Determine how many gallons of gasoline are carried to vehicles with empty fuel tanks each year in the United States so that the vehicle can be driven to the nearest gas sta-tion. Do this without reference to any other material, such as the Internet or reference books.
Estimations
In almost all cases, the first step is to estimate unknown pieces of information that are not avail-able. In general, there are numerous paths to a solution, and different people may arrive at different answers. Often, the answer arrived at is only accurate to within an order of magnitude (a factor of 10) or less. Nonetheless, such problems can provide valuable insight not only into the problem itself but also into the nature of problem solving in general.
Remember, these are estimates, not accurate values. Someone else making these estimates might make different assumptions.
■ Number of people in the United States: 500,000,000 persons.
■ Fraction of people in the United States that drive: drivers per 10 persons.
■ Average times a person runs out of gas per year: one “out of gas” per 4 years per driver.
■ Fraction of “out of gas” incidents in which gas is brought to the car (rather than pushing or towing the car to a station): 24 “bring gas to car” per 25 “out of gas”.
EXAMPLE 5-1
■ Average amount of gas carried to car: 1.5 gallons per “bring gas to car”.
Calculation
This is where you combine your estimates to arrive at a solution. In the process, you may realize that you need further information to complete the computation.
Drivers in United States = 15 * 108 people2 a 7 drivers
10 peopleb = 3.5 * 108 drivers
Number out of gas per year = 13.5 * 108 drivers2 a 1 out of gas (4 years)(1 driver)b
= 8.75 * 107out of gas
year
Number bring gas to car per year = a8.75 * 107out of gas
year b a24 bring to car 25 out of gas b
= 8.4 * 107bring gas to car
year
Amount of gas to cars = a8.4 * 107 bring gas to car
year b a1.5 gallons of gas bring gas to car b
= 1.26 * 108gallons
year
Thus, we have estimated that about 125 million gallons of gas are taken to “out of fuel”
vehicles each year in the United States. It would be perfectly valid to give the answer as
“about 100 million” gallons, since we probably have only about one significant digit worth of confidence in our results, if that.
A few things to note about this solution:
■ “Units” were used on all numerical values, although some of these units were somewhat contrived (e.g., “out of gas”) to meet the needs of the problem. Keeping track of the units is critical to obtaining correct answers and will be highly emphasized, not only in this text but also throughout your engineering education and career.
■ The units combine and cancel according to the regular algebraic rules. For example, in the first computation, the unit “persons” appeared in both the numerator and the denominator, and thus canceled, leaving “drivers.”
■ Rather than the computation being combined into one huge string of computations, it was broken into smaller pieces, with the results from one step used to compute the next step. This is certainly not an immutable rule, but for long computations it reduces care-less errors and makes it easier to understand the overall flow of the problem.
Plastic resin (shown in the picture, in pellet form) is used in many types of manufacturing methods including injec-tion molding, extrusion, and blow molding. Injecinjec-tion molding describes the process by which resin is melted and “injected” into a closed mold, then cooled forming the final part. Extrusion is a continuous process by which resin is melted and pushed through an open mold to cre-ate shapes like a pipe or rod.
Most cars today use plastic fuel tanks made by blow molding. In blow molding, the resin is melted and pushed through an extrusion head that forms the plastic into a hollow shape. The shape is then pressurized with air and cooled in a mold to form the part. Most fuel tanks are formed with six layers of various plastic to increase toughness and eliminate permeation of the fuel through the tank.
Photo credit: E. Stephan
5.1 GENERAL HINTS FOR ESTIMATION
As you gain more knowledge and experience, the types of problems you can estimate will become more complicated. Here we give you a few hints about making estimates.
■ Try to determine the accuracy required.
• Is order of magnitude enough? Is {25%?
• What level of accuracy is needed to calculate a satellite trajectory?
• What level of accuracy is needed to determine the amount of paint needed to paint a specified classroom?
The term “orders of magnitude” is often used when comparing things of very different scales, such as a small rock and a planet. By far the most common usage refers to fac-tors of 10; for example, three orders of magnitude refer to a difference in scale of 10 3 = 1,000. If we wanted to consider the order of magnitude between 10,000,000 and 1,000, we would calculate the logarithm of each value (log(10,000,000) = 7 and log(1,000) = 3), thus there are 7 - 3 = 4 orders of magnitude difference between 10,000,000 and 1,000.
LEARN TO: Determine how much accuracy is needed in a particular situation Identify the important variables affecting an estimate
■ Remember that a “ballpark” value is often good enough for an input parameter.
• What is the square footage of a typical house?
• What is the maximum high temperature to expect in Dallas, Texas, in July?
• What is the typical velocity of a car on the highway?
■ Always ask yourself if it is better to err on the high side or the low side.
• Safety and practical considerations. Will a higher or lower estimate result in a safer or more reliable result?
■ If estimating the weight a bridge can support, it is better to err on the low side, so that the actual load it can safely carry is greater than the estimate.
■ For the bridge mentioned above, if estimating the load a single beam needs to support, it is better to err on the high side, thus giving a stronger beam than necessary. Be sure you understand the difference between these two points.
• Estimate improvement. Can the errors cancel each other?
■ If estimating the product of two numbers, if one of the terms is rounded low, the other should be rounded high to counteract for the lower term.
■ If estimating a quotient, if you round the numerator term on the low side, should the denominator term be rounded low or high?
■ Do not get bogged down with second-order or minor effects.
• If estimating the mass of air in the classroom, do you need to correct for the pres-ence of furniture?
• In most instances, can the effect of temperature on the density of water be neglected?
The best way to develop your ability to estimate is through experience. An expe-rienced painter can more easily estimate how much paint is needed to repaint a room because experience will have taught the painter such things as how many coats of one paint color it will take to paint over another, how different paint brands differ in their coverage, and how to estimate surface area quickly. In Outliers: The Story of Success, Malcolm Gladwell provides examples from diverse career pathways that demonstrate 10,000 hours of practice are required to develop world-class expertise in any area.
Fortunately for aspiring engineers, much of this expertise can be developed starting at a young age and outside of formal schooling. For example, how many hours have you spent observing the effects of gravity? Of course, some important engineering con-cepts stem from phenomena that are not so easily observed, and some lend themselves to misinterpretation. As a result, it helps to have a systematic approach to developing estimates—particularly where we have less experience to guide us. Estimating an approximate answer of a calculation including known quantities is a valuable skill—
such as approximating the square root of 50 as about 7, approximating the value of pi as 3 for quick estimates, etc. These mathematical approximations, however, assume that you have all the numbers to begin with, and that you can use shortcuts to estimate the precisely calculated answer to save time or as a check against your more carefully calculated answer. Estimation is discussed here in a broader sense—estimating quanti-ties that cannot be known, are complicated to measure, or are otherwise inconvenient to obtain. It is in these cases that the following strategies are recommended.
The Windows interface estimates the time needed to copy files. The estimate is dynamic and appears to be based on the total number of files and the assumption that each file will take the same amount of time to copy. As a result, when large files are copied, the estimate will increase—sometimes significantly. Similarly, as a large num-ber of small files are copied, the estimate will decrease rapidly. A better estimation algorithm might be based on the percentage of the total file size.
Estimate the size of an acre and a hectare of land using analogy.
American football field—playing area is 300 feet by 160 feet = approximately 50,000 square feet. An acre is 43,560 square feet. Using this data, we have a better sense of how much land an acre is—about 90% of the size of the playing area of an American football field. Soccer fields are larger, but vary in size. The largest soccer field that satisfies interna-tional guidelines would be about 2 acres.
A hectare, or 10,000 square meters, is equivalent to 108,000 square feet and is much larger than an acre—about the maximum size of the pitch in international rugby competition.
EXAMPLE 5-3
5.3 ESTIMATION BY AGGREGATION
Another useful strategy is to estimate the quantity of something by adding up an estimate of its parts. This can involve multiplication in the case of a number of simi-larly sized parts, such as estimating the size of a tile by comparing it to your foot (estimation by analogy), counting the number of floor tiles across a room, and multi-plying to estimate the total length of the room. In other cases, aggregation may involve adding together parts that are estimated by separate methods. For example, to Estimate the size of a laptop computer using analogy.
Laptop computers come in different sizes, but it is not difficult to estimate the size of a particular laptop. Laptops were first called “notebook” computers—a good starting esti-mate would be to compare the particular laptop to notebook paper, which is 8.5 inches by 11 inches in the United States.
EXAMPLE 5-2
5.2 ESTIMATION BY ANALOGY
One useful strategy for estimating a quantity is by comparison to something else we have measured previously or otherwise know the dimension of. The best way to pre-pare for this approach to estimation is to learn a large number of comparison measures for each type of quantity you might wish to estimate. Each of these comparison mea-sures becomes an anchor point on that scale of measurement. This book provides some scale anchors for various measurable quantities—particularly in the case of power and energy, concepts with which many people struggle.
LEARN TO: Recognize how to use aggregation as a tool for estimation LEARN TO: Recognize how to use analogy as a tool for estimation
5.4 ESTIMATION BY UPPER AND LOWER BOUNDS
An important part of estimating is keeping track of whether your estimate is high or low. In the earlier example of estimating the volume of a two-scoop ice cream cone, we would have over-estimated, because one of the scoops of ice cream is pressed inside the cone. The effect of pressing the scoops together is not that important, because the same amount of ice cream is still there, but if the scoop is pressed into an ellipsoid, it may be difficult to estimate the original radius of the scoop.
Engineers frequently make “conservative” estimates, which consider the “worst- case” scenario. Depending on the situation, the worst case may be a lower limit (such as estimating the strength of a structure) or an upper limit (such as estimating how much material is needed for a project).
If you are to estimate how many gallons of paint are needed to paint the room you are in, what assumptions will you need to make? Where will you need to make assump-tions to ensure that you have enough paint without running out?
In estimating the wall area, you should round up the length and height.
Noting that paint (for large jobs) is sold in 5-gallon pails, you will want to round your final estimate to the next whole 5-gallon pail.
Close estimates allow for subtracting 21 square feet per doorway (if the doors are not being painted the same color). In making a rough estimate, if there are not a lot of door-ways, it would be conservative to leave in the door area.
EXAMPLE 5-5
Estimate by aggregation the amount of money students at your school spend on pizza each year.
Ask students around you how often they purchase a pizza and how much it costs;
Convert this estimate into a cost per week;
Multiply your estimate by the number of weeks in an academic year;
Multiple that result by the number of students at your school.
EXAMPLE 5-4
estimate the volume of a two-scoop ice cream cone, you might estimate the volume of the cone and then separately estimate the volume of each scoop assuming they are each spheres.
LEARN TO: Understand how upper and lower bounds can guide estimation
5.6 SIGNIFICANT FIGURES
Significant figures or “sig figs” are the digits considered reliable as a result of measure-ment or calculation. This is not to be confused with the number of digits or decimal places. The number of decimal places is simply the number of digits to the right of the decimal point. Example 5-8 illustrates these two concepts.
A large sample of sunflower seeds is collected and their lengths are measured. Using that information, estimate the length of the longest sunflower seed you are likely to find if you measure one billion seeds.
Given a large sample, its average and standard deviation can be calculated. Assuming that the length of sunflower seeds is normally distributed, the one-in-a-billion largest sunflower seed would be expected to be six standard deviations greater than the sample average.
EXAMPLE 5-7
You would like to enjoy a bowl of peas, but they are too hot to eat. Spreading them out on a plate allows them to cool faster. Describe why this happens and devise a model of how much faster the peas on the plate will cool. Mice have a harder time keeping warm compared to elephants. Explain how this is related to the bowl of peas. How does this relate to the fact that smaller animals have higher heart rates? Canaries and humming-birds can have heart rates of 1,200 beats per minute, whereas human heart rates should not exceed 150 beats per minute even during exercise.
The peas cool faster when spread out because of the increase in surface area. The ratio of surface area (proportional to cooling) to volume (proportional to the heat capacity for a particular substance) is therefore important. Similarly, smaller animals have a harder time staying warm because they have a higher ratio of surface area to volume. The higher heart rate is needed to keep their bodies warm. This also relates to why smaller animals consume a much larger amount of food compared to their body mass.
EXAMPLE 5-6
5.5 ESTIMATION USING MODELING
In cases that are more complicated or where a more precise estimate is required, math-ematical models and statistics might be used. Sometimes dimensionless quantities are useful for characterizing systems, sometimes modeling the relationship of a small num-ber of variables is needed, and at other times, extrapolating even a single variable from available data is all that is needed to make an estimate.
LEARN TO: Understand how models can guide estimation
LEARN TO: Defi ne signifi cant fi gures within a value
Understand how to determine the number of signifi cant fi gures in calculations