Exercise 3. A piece of tubing is 0.566 cm long. How many μm does this correspond to? How many km?

Answer: 5.66 x 10³ μm and 5.66 x 10^-6 km

Solution

When converting from cm to μm. Two equivalences are:

\(100 \;cm \;= \;1 \;m \quad and \quad 10^{-6}\; m\; =\; 1 \;μm\)
 
We can write two sets of conversion factors from the 2 equivalents

\(\frac{100\;cm}{1\;m}\quad or \quad \frac{1\;m}{100\;cm}\quad and \quad \frac{10^{-6}\;m}{1\;μm}\quad or \quad \frac{1\;μm}{10^{-6}\;m}\)
 
We are going from \(cm\;\rightarrow\;m\rightarrow\;μm\)
The above is our roadmap. Notice, there are two arrows. We have one equivalence for each arrow. Going from cm to m, we use the equivalence 100 cm = 1 m. Going from m to μm, we use the equivalence, 10^-6 m = 1 μm. Again, you can use different equivalences to come up with conversion factors. As long as your equivalences are correct, you will obtain the correct answer. Now, we are ready to set up our work. Start with the given, 0.566 cm. Look at the roadmap, and use the conversion factor that will cancel out cm. This will get us to m. We then refer back to the roadmap, and choose the conversion factor that will cancel out m and leave us with μm.

\(\require{cancel}0.566\cancel{cm}\times\frac{1\cancel{m}}{100\cancel{cm}}\times\frac{1\;μm}{10^{-6}\cancel{m}}=\;5.66\times10^{3}\;μm\)
 
When converting from cm to km. Two equivalences are:
\(100 \;cm \;= \;1 \;m \quad and \quad 1000\; m\; =\; 1 \;km\)
 
We can write two sets of conversion factors:

\(\frac{100\;cm}{1\;m}\quad or \quad \frac{1\;m}{100\;cm}\quad and \quad \frac{1000\;m}{1\;km}\quad or \quad \frac{1\;km}{1000\;m}\)
 
We are going from \(cm\;\rightarrow\;m\rightarrow\;km\)

The above is our roadmap. Notice, there are two arrows. We need one equivalence for each arrow. Going from cm to m, we use the equivalence 100 cm = 1 m. Going from m to km, we use the equivalence, 1000 m = 1 km. Again, you can use different equivalences to come up with conversion factors. As long as your equivalences are correct, you will obtain the correct answer. Now, we are ready to set up our work. Start with the given, 0.566 cm. Look at the roadmap, and use the conversion factor that will cancel out cm. This will get us to m. We then refer back to the roadmap, and choose the conversion factor that will cancel out m and leave us with km.

\(\require{cancel}0.566\cancel{cm}\times\frac{1\cancel{m}}{100\cancel{cm}}\times\frac{1\;km}{1000\cancel{m}}=\;5.66\times10^{-6}\;km\)
 
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