Question 2.14: Solving Multistep Conversion Problems Involving Units Raised......
Solving Multistep Conversion Problems Involving Units Raised to a Power
The average annual per person crude oil consumption in the United States is 15,615 dm^3. What is this value in cubic inches?
| SORT
You are given a volume in cubic decimeters and asked to convert it to cubic inches. |
GIVEN: 15,615 dm^3
FIND: in.^3 |
| STRATEGIZE
Build a solution map beginning with dm^3 and ending with in.^3 You must cube each of the conversion factors because the quantities involve cubic units. |
SOLUTION MAP
dm^3 \underset{\frac{(0.1 m)^3}{(1 dm)^3} }{\longrightarrow } m^3 \underset{\frac{(1 cm)^3}{(0.01 m)^3} }{\longrightarrow }cm^3 \underset{\frac{(1 in.)^3}{(2.54 cm)^3} }{\longrightarrow } in.^3
RELATIONSHIPS USED 1 dm = 0.1 m (from Table 2.2) 1 cm = 0.01 m (from Table 2.2) 2.54 cm = 1 in. (from Table 2.3) |
| SOLVE
Follow the solution map to solve the problem. Begin with the given value in dm^3 and multiply by the string of conversion factors to arrive at in.^3 Be sure to cube each conversion factor as you carry out the calculation. Round the answer to five significant figures to reflect the five significant figures in the least precisely known quantity (15,615 dm^3). The conversion factors are all exact and therefore do not limit the number of significant figures. |
SOLUTION
15,615 \cancel{dm^3} \times \frac{(0.1 \cancel{m})^3}{(1 \cancel{dm})^3} \times \frac{(1 \cancel{cm})^3}{(0.01 \cancel{m})^3} \times \frac{(1 in.)3}{(2.54 \cancel{cm})^3}
= 9.5289 \times 10^5 in.^3 |
| CHECK
Check your answer. Are the units correct? Does the answer make physical sense? |
The units of the answer are correct, and the magnitude makes sense. A cubic inch is smaller than a cubic decimeter, so the value in cubic inches should be larger than the value in cubic decimeters. |
Table 2.2
| SI Prefix Multipliers | ||||
| Prefix | Symbol | Meaning | Multiplier | |
| tera- | T | trillion | 1,000,000,000,000 | (10^{12}) |
| giga- | G | billion | 1,000,000,000 | (10^{9}) |
| mega- | M | million | 1,000,000 | (10^{6}) |
| kilo- | k | thousand | 1,000 | (10^{3}) |
| hecto- | h | hundred | 100 | 10^{2} |
| deca- | da | ten | 10 | 10^{1} |
| deci- | d | tenth | 0.1 | (10^{-1}) |
| centi- | c | hundredth | 0.01 | (10^{-2}) |
| milli- | m | thousandth | 0.001 | (10^{-3}) |
| micro- | µ | millionth | 0.000001 | (10^{-6}) |
| nano- | n | billionth | 0.000000001 | (10^{-9}) |
| pico- | p | trillionth | 0.000000000001 | (10^{-12}) |
| femto- | f | quadrillionth | 0.000000000000001 | (10^{-15}) |
Table 2.3
| Some Common Units and Their Equivalents |
| Length |
| 1 kilometer (km) = 0.6214 mile (mi) |
| 1 meter (m) = 39.37 inches (in.)
= 1.094 yards (yd) |
| 1 foot (ft) = 30.48 centimeters (cm) |
| 1 inch (in.) = 2.54 centimeters (cm) (exact) |
| Mass |
| 1 kilogram (kg) = 2.205 pounds (lb) |
| 1 pound (lb) = 453.59 grams (g) |
| 1 ounce (oz) = 28.35 grams (g) |
| Volume |
| 1 liter (L) = 1000 milliliters (mL)
= 1000 cubic centimeters (cm^3) |
| 1 liter (L) = 1.057 quarts (qt) |
| 1 U.S. gallon (gal) = 3.785 liters (L) |