Convert by using unit factors Show setups at right: (a) 17 m dm 11 / 0. | (b) 1.5 kg 8 (c) 14 cm (d) 1.7 pL (e) 9.25 m? cm? (0 3.4 km: m? 3, 4n
The Correct Answer and Explanation is:
Here are the correct answers for the unit conversions:
(a) 17 m = 170 dm
(b) 1.5 kg = 1500 g
(c) 14 cm = 0.14 m
(d) 1.7 µL = 0.0000017 L
(e) 9.25 m² = 92,500 cm²
(f) 3.4 km³ = 3,400,000,000 m³
Explanation
This problem requires converting measurements using unit factors, a method also known as dimensional analysis. This technique involves multiplying the original quantity by a conversion factor—a fraction that equals one—to change the units without altering the value. A unit factor is created from an equality, such as 1 meter = 10 decimeters. This can be written as the fraction (10 dm / 1 m) or (1 m / 10 dm), both of which are equal to one. You choose the fraction that allows the original unit to cancel out, leaving the desired unit.
(a) 17 m to dm: Since 1 meter equals 10 decimeters, the unit factor needed is (10 dm / 1 m) to cancel out meters.
Setup: 17 m × (10 dm / 1 m) = 170 dm
(b) 1.5 kg to g: The prefix “kilo” means 1000, so 1 kilogram equals 1000 grams.
Setup: 1.5 kg × (1000 g / 1 kg) = 1500 g
(c) 14 cm to m: The prefix “centi” means 1/100, so 100 centimeters are in 1 meter. To cancel centimeters, we place it in the denominator.
Setup: 14 cm × (1 m / 100 cm) = 0.14 m
(d) 1.7 µL to L: The prefix “micro” (µ) means one-millionth, so 1 liter contains 1,000,000 microliters.
Setup: 1.7 µL × (1 L / 1,000,000 µL) = 0.0000017 L (or 1.7 x 10⁻⁶ L)
(e) 9.25 m² to cm²: For area conversions, the linear unit factor must be squared. Since 1 m = 100 cm, we square the entire relationship: (1 m)² = (100 cm)², which gives 1 m² = 10,000 cm².
Setup: 9.25 m² × (10,000 cm² / 1 m²) = 92,500 cm²
(f) 3.4 km³ to m³: For volume, the linear unit factor is cubed. Since 1 km = 1000 m, we cube the relationship: (1 km)³ = (1000 m)³, which gives 1 km³ = 1,000,000,000 m³.
Setup: 3.4 km³ × (1,000,000,000 m³ / 1 km³) = 3,400,000,000 m³
