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5.8: Problem-Solving Strategies: Dimensional Analysis

Difficulty Level: At Grade Created by: CK-12
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Real-world information is given in dimensions, or the units in which the value is measured. For example, the following are all examples of dimensions.

  • Inches
  • Feet
  • Liters
  • Micrograms
  • Acres
  • Hours
  • Pounds
  • Students

Analyzing dimensions can help you solve problems in travel, astronomy, physics, engineering, forensics, and quality. Solving problems by converting dimensions or canceling dimensions is the focus of this lesson.

Consider the distance formula \begin{align*}distance=rate \ \cdot \ time\end{align*}distance=rate  time. This formula can be rewritten for rate. \begin{align*}rate=\frac{distance}{time}\end{align*}rate=distancetime. If distance is measured in kilometers, and time is measured in hours, the rate would have the dimensions \begin{align*}\frac{kilometers}{hours}\end{align*}kilometershours.

You can treat dimensions as variables. Identical units can divide out, or cancel. For example, \begin{align*}\frac{kilometers}{hour} \cdot hour \rightarrow \frac{kilometers}{\cancel{hour}} \cdot \cancel{hour} \rightarrow kilometers\end{align*}kilometershourhourkilometershourhourkilometers.

Sometimes the units will not divide out. In this case, a conversion factor is needed.

Example: Convert \begin{align*}\frac{35 \ kilometers}{hour}\end{align*}35 kilometershour to meters.

Solution: Since kilometers \begin{align*}\neq\end{align*} meters, you need to convert kilometers to meters to get the answer. You know there are 1,000 meters in a kilometer. Therefore, you will need to multiply the original dimension by this factor.

\begin{align*}\frac{35 \ kilometers}{hour} \cdot \frac{1000 \ meters}{1 \ kilometer} \cdot hour \rightarrow \frac{35 \ \cancel{kilometers}}{\cancel{hour}} \cdot \frac{1000 \ meters}{1 \ \cancel{kilometer}} \cdot \cancel{hour} & = 35(1000)meters.\\ \frac{35\ kilometers}{hour}& =35,000 \ meters\end{align*}35 kilometershour1000 meters1 kilometerhour35 kilometershour1000 meters1 kilometerhour35 kilometershour=35(1000)meters.=35,000 meters

The process of using units or dimensions to help solve a problem is called dimensional analysis. It is very useful in chemistry and travel, as shown in the examples below.

Example: How many seconds are in a month?

Solution: This situation can be solved easily using multiplication. However, the process you use when multiplying the values together is an example of dimensional analysis.

Begin with what you know:

  • 60 seconds in one minute
  • 60 minutes in one hour
  • 24 hours in one day
  • Approximately 30 days in one month

Now write the expression to convert the seconds in one minute to one month.

\begin{align*}\frac{60 \ seconds}{1 \ minute} \cdot \frac{60 \ minutes}{1 \ hour} \cdot \frac{24 \ hours}{1 \ day} \cdot \frac{30 \ days}{1 \ month}\end{align*}60 seconds1 minute60 minutes1 hour24 hours1 day30 days1 month

Identical units cross-cancel.

\begin{align*}\frac{60 \ seconds}{1 \ \cancel{minute}} \cdot \frac{60 \ \cancel{minutes}}{1 \ \cancel{hour}} \cdot \frac{24 \ \cancel{hours}}{1 \ \cancel{day}} \cdot \frac{30 \ \cancel{days}}{1 \ month}\end{align*}60 seconds1 minute60 minutes1 hour24 hours1 day30 days1 month

Multiply the fractions together.

\begin{align*}\frac{60 \cdot 60 \cdot 24 \cdot 30 \ seconds}{1 \cdot 1 \cdot 1 \cdot 1 \ month}=2,592,000\frac{seconds}{month}\end{align*}60602430 seconds1111 month=2,592,000secondsmonth

Example 1: How many grams are in 5 pounds?

Solution: Begin by writing all the conversions you know related to this situation.

\begin{align*}1 \ gram & \approx 0.0353 \ ounces\\ 16 \ ounces & = 1 \ pound\end{align*}1 gram16 ounces0.0353 ounces=1 pound

Write your dimensional analysis.

\begin{align*}5 \ pounds \cdot \frac{16 \ ounces}{1 \ pound} \cdot \frac{1 \ gram}{0.0353 \ ounce}\end{align*}5 pounds16 ounces1 pound1 gram0.0353 ounce

Cross-cancel identical units and multiply.

\begin{align*}5 \ \cancel{pounds} \cdot \frac{16 \ \cancel{ounces}}{1 \ \cancel{pound}} \cdot \frac{1 \ gram}{0.0353 \ \cancel{ounce}}=2226.29 \ grams\end{align*}5 pounds16 ounces1 pound1 gram0.0353 ounce=2226.29 grams

A long list of conversion factors can be found at this website.

Practice Set

  1. True or false? Dimensional analysis is the study of space and time.
  2. By using dimensional analysis, what happens to identical units that appear diagonally in the multiplication of fractions?
  3. How many feet are in a mile?
  4. How many inches are in a mile?
  5. How many seconds are in a day?
  6. How many seconds are in a year?
  7. How many feet are in a furlong?
  8. How many inches are in 100 yards (one football field)?
  9. How many centimeters are in 5 inches?
  10. How many meters are between first and second base (90 feet)?
  11. How many meters are in 16 yards?
  12. How many cups are in 6 liters?
  13. How many cubic inches make up one ounce?
  14. How many milliliters make up 8 ounces?
  15. How many grams are in 100 pounds?
  16. An allergy pill contains 25 mg of Diphenhydramine. If \begin{align*}1 \ gram=15.432 \ grains\end{align*}1 gram=15.432 grains, how many grains of this medication is in the allergy pill?
  17. A healthy individual’s heart beats about 68 times per minute. How many beats per hour is this?
  18. You live 6.2 miles from the grocery store. How many fathoms is this? (\begin{align*}6 \ feet=1 \ fathom\end{align*}6 feet=1 fathom)
  19. The cost of gas in England is 96.4 pound sterling/liter. How much is this in U.S. dollars/gallon? (\begin{align*}3.875\ litres=1 \ gallon\end{align*}3.875 litres=1 gallon and 1.47 US \begin{align*}\$=1\end{align*}$=1 pound sterling)
  20. Light travels \begin{align*}\frac{186,000 \ miles}{second}\end{align*}186,000 milessecond. How long is one light year?
  21. Another way to describe light years is in astronomical units. If 1 \begin{align*}light \ year=63,240 \ AU\end{align*}light year=63,240 AU (astronomical units), how far in AUs is Alpha Centauri, which is 4.32 light years from the Earth?
  22. How many square feet is 16 acres?
  23. A person weighs 264 pounds. How many kilograms is this weight?
  24. A car is traveling 65 miles/hour and crosses into Canada. What is this speed in km/hr?
  25. A large soda cup holds 32 ounces. What is this capacity in cubic inches?
  26. A space shuttle travels 28,000 mph. What is this distance in feet/second?
  27. How many hours are in a fortnight (two weeks)?
  28. How many fortnights (two-week periods) are in 2 years?
  29. A semi truck weighs 32,000 pounds empty. How many tons is this weight?
  30. Which has the greatest volume: a 2-liter bottle of soda, one gallon of water, or 10 pints of human blood?

Mixed Review

  1. Solve for \begin{align*}x: -2x+8=8(1-4x)\end{align*}.
  2. Simplify: \begin{align*}3-2(5-8h)+13h \cdot 3\end{align*}.
  3. Find the difference: \begin{align*}-26.375-\left (-14 \frac{1}{8}\right )\end{align*}.
  4. Find the product: \begin{align*}-2\frac{3}{7} \cdot \frac{9}{10}\end{align*}.
  5. Simplify: \begin{align*}\sqrt{80}\end{align*}.
  6. Is \begin{align*}5.\bar{5}\end{align*} an irrational number? Explain your answer.

Use the relation given for the following questions: \begin{align*}\left \{(0,8),(1,4),(2,2),(3,1),\left(4,\frac{1}{2}\right ),\left (5,\frac{1}{4}\right )\right \}\end{align*}.

  1. State the domain.
  2. State the range.
  3. Is this relation a function? Explain your answer.
  4. What seems to be the pattern in this relation?

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