# 1.6 Mathematical treatment of measurement results  (Page 3/9)

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## Computing quantities from measurement results and known mathematical relations

(a) What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?

(b) If gasoline costs $3.80 per gallon, what was the fuel cost for this trip? ## Solution (a) We first convert distance from kilometers to miles: $\text{1250 km}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{0.62137 mi}}{\text{1 km}}\phantom{\rule{0.2em}{0ex}}=\text{777 mi}$ and then convert volume from liters to gallons: $213\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1.0567}\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1 gal}}{4\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}\phantom{\rule{0.2em}{0ex}}=\text{56.3 gal}$ Then, $\text{(average) mileage}=\phantom{\rule{0.2em}{0ex}}\frac{\text{777 mi}}{\text{56.3 gal}}\phantom{\rule{0.2em}{0ex}}=\text{13.8 miles/gallon}=\text{13.8 mpg}$ Alternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows: $\frac{1250\phantom{\rule{0.2em}{0ex}}\overline{)\text{km}}}{213\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{0.62137 mi}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{km}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}}{\text{1.0567}\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{4\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}{\text{1 gal}}=\text{13.8 mpg}$ (b) Using the previously calculated volume in gallons, we find: $\text{56.3 gal}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{3.80}}{\text{1 gal}}=\text{214}$ ## Check your learning A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits). (a) What (average) fuel economy, in miles per gallon, did the Prius get during this trip? (b) If gasoline costs$3.90 per gallon, what was the fuel cost for this trip?

(a) 51 mpg; (b) \$62

## Conversion of temperature units

We use the word temperature to refer to the hotness or coldness of a substance. One way we measure a change in temperature is to use the fact that most substances expand when their temperature increases and contract when their temperature decreases. The mercury or alcohol in a common glass thermometer changes its volume as the temperature changes. Because the volume of the liquid changes more than the volume of the glass, we can see the liquid expand when it gets warmer and contract when it gets cooler.

To mark a scale on a thermometer, we need a set of reference values: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On the Celsius scale, 0 °C is defined as the freezing temperature of water and 100 °C as the boiling temperature of water. The space between the two temperatures is divided into 100 equal intervals, which we call degrees. On the Fahrenheit    scale, the freezing point of water is defined as 32 °F and the boiling temperature as 212 °F. The space between these two points on a Fahrenheit thermometer is divided into 180 equal parts (degrees).

Defining the Celsius and Fahrenheit temperature scales as described in the previous paragraph results in a slightly more complex relationship between temperature values on these two scales than for different units of measure for other properties. Most measurement units for a given property are directly proportional to one another (y = mx). Using familiar length units as one example:

$\text{length in feet}=\left(\frac{\text{1 ft}}{\text{12 in.}}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{length in inches}$

where y = length in feet, x = length in inches, and the proportionality constant, m, is the conversion factor. The Celsius and Fahrenheit temperature scales, however, do not share a common zero point, and so the relationship between these two scales is a linear one rather than a proportional one (y = mx + b). Consequently, converting a temperature from one of these scales into the other requires more than simple multiplication by a conversion factor, m, it also must take into account differences in the scales’ zero points (b).

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