What is the shortest of the following? (a) 10^4 in. (b) 10^4 m (c) 10^3 ft (d) 0.1 mi

Steps to Solve

1. Convert all lengths to inches. To compare the lengths effectively, we convert all options to inches.

2. Convert 10^4 meters to inches. Use the conversion factor 1 meter = 39.37 inches. $$ 10^4 \text{ m} = 10^4 \text{ m} \times \frac{39.37 \text{ in}}{1 \text{ m}} = 39.37 \times 10^4 \text{ in} = 3.937 \times 10^5 \text{ in} $$

3. Convert 10^3 feet to inches. Use the conversion factor 1 foot = 12 inches. $$ 10^3 \text{ ft} = 10^3 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} = 12 \times 10^3 \text{ in} = 1.2 \times 10^4 \text{ in} $$

4. Convert 0.1 miles to inches. Use the conversion factor 1 mile = 5280 feet and 1 foot = 12 inches. $$ 0.1 \text{ mi} = 0.1 \text{ mi} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{12 \text{ in}}{1 \text{ ft}} = 0.1 \times 5280 \times 12 \text{ in} = 6336 \text{ in} = 6.336 \times 10^3 \text{ in} $$

5. Compare the lengths in inches. We have: (a) $10^4$ in $= 10000$ in (b) $10^4$ m $= 3.937 \times 10^5$ in $= 393700$ in (c) $10^3$ ft $= 1.2 \times 10^4$ in $= 12000$ in (d) $0.1$ mi $= 6.336 \times 10^3$ in $= 6336$ in

Comparing the values, we see that 0.1 miles is the shortest.