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Difficulty: MediumScientific Notation and Unit Conversions

During an astrophysics study, the density of a stellar nebula's core is determined to be 4.5×10−124.5 \times 10^{-12} grams per cubic centimeter (g/cm3\text{g/cm}^3). Which of the following is equivalent to this density expressed in kilograms per cubic meter (kg/m3\text{kg/m}^3)?

  1. A
    4.5×10−15 kg/m34.5 \times 10^{-15}\text{ kg/m}^3
  2. 4.5×10−9 kg/m34.5 \times 10^{-9}\text{ kg/m}^3Answer
  3. C
    4.5×10−12 kg/m34.5 \times 10^{-12}\text{ kg/m}^3
  4. D
    4.5×10−6 kg/m34.5 \times 10^{-6}\text{ kg/m}^3

Answer

4.5×10−9 kg/m34.5 \times 10^{-9}\text{ kg/m}^3
The correct answer is 4.5×10−9 kg/m34.5 \times 10^{-9}\text{ kg/m}^3. To convert density from grams per cubic centimeter (g/cm3\text{g/cm}^3) to kilograms per cubic meter (kg/m3\text{kg/m}^3), we convert the numerator (1 g=10−3 kg1\text{ g} = 10^{-3}\text{ kg}) and the denominator (1 cm3=10−6 m31\text{ cm}^3 = 10^{-6}\text{ m}^3). This gives the conversion: 1 g/cm3=10−3 kg10−6 m3=103 kg/m31\text{ g/cm}^3 = \frac{10^{-3}\text{ kg}}{10^{-6}\text{ m}^3} = 10^3\text{ kg/m}^3. Multiplying the initial measurement by this factor gives: (4.5×10−12)×103=4.5×10−9 kg/m3(4.5 \times 10^{-12}) \times 10^3 = 4.5 \times 10^{-9}\text{ kg/m}^3.

Step-by-Step Solution

1
Convert the mass unit from grams (g\text{g}) to kilograms (kg\text{kg}).
1 g=10−3 kg1\text{ g} = 10^{-3}\text{ kg}
Since 1 kg=103 g1\text{ kg} = 10^3\text{ g}, converting from a smaller unit (g) to a larger unit (kg) requires multiplying by 10−310^{-3}.
2
Convert the volume unit in the denominator from cubic centimeters (cm3\text{cm}^3) to cubic meters (m3\text{m}^3).
1 cm3=(10−2 m)3=10−6 m31\text{ cm}^3 = (10^{-2}\text{ m})^3 = 10^{-6}\text{ m}^3
Since 1 m=102 cm1\text{ m} = 10^2\text{ cm}, cubing both sides of the conversion factor gives 1 m3=106 cm31\text{ m}^3 = 10^6\text{ cm}^3, or 1 cm3=10−6 m31\text{ cm}^3 = 10^{-6}\text{ m}^3.
3
Combine the conversion factors to find the overall density conversion factor from g/cm3\text{g/cm}^3 to kg/m3\text{kg/m}^3.
1 g1 cm3=10−3 kg10−6 m3=103 kg/m3\frac{1\text{ g}}{1\text{ cm}^3} = \frac{10^{-3}\text{ kg}}{10^{-6}\text{ m}^3} = 10^3\text{ kg/m}^3
Dividing the mass conversion factor by the volume conversion factor yields the combined multiplier for density.
4
Multiply the given core density by the combined conversion factor.
(4.5×10−12)×103=4.5×10−9(4.5 \times 10^{-12}) \times 10^3 = 4.5 \times 10^{-9}
Applying the combined factor converts the value to the desired units.

Key Concept

Unit conversion involving compound units and powers of ten

Alternative Method

We can set up dimensional analysis: 4.5×10−12 g/cm3×(1 kg1000 g)×(100 cm1 m)3=4.5×10−12×10−3×106=4.5×10−9 kg/m34.5 \times 10^{-12} \text{ g/cm}^3 \times \left(\frac{1\text{ kg}}{1000\text{ g}}\right) \times \left(\frac{100\text{ cm}}{1\text{ m}}\right)^3 = 4.5 \times 10^{-12} \times 10^{-3} \times 10^6 = 4.5 \times 10^{-9} \text{ kg/m}^3.
Estimated Time:1m 30s
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Topics

Scientific Notation and Unit Conversions

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