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Difficulty: MediumScientific Notation and Unit Conversions

An environmental microbiologist studying soil ecology estimates the average volume of a single soil bacterium to be 2.0×10−122.0 \times 10^{-12} cubic centimeters (cm3\text{cm}^3). What is the average volume of this bacterium expressed in cubic micrometers (μm3\mu\text{m}^3)?

  1. A
    2.0×10−24 μm32.0 \times 10^{-24} \text{ } \mu\text{m}^3
  2. B
    2.0×10−8 μm32.0 \times 10^{-8} \text{ } \mu\text{m}^3
  3. C
    2.0×10−6 μm32.0 \times 10^{-6} \text{ } \mu\text{m}^3
  4. 2.0×100 μm32.0 \times 10^0 \text{ } \mu\text{m}^3Answer

Answer

2.0×100 μm32.0 \times 10^0 \text{ } \mu\text{m}^3
The correct answer is 2.0×100 μm32.0 \times 10^0 \text{ } \mu\text{m}^3. To convert cubic centimeters to cubic micrometers, we first find that 1 cm=104 μm1 \text{ cm} = 10^4 \text{ } \mu\text{m}. Cubing both sides of this linear relationship yields (1 cm)3=(104 μm)3=1012 μm3(1 \text{ cm})^3 = (10^4 \text{ } \mu\text{m})^3 = 10^{12} \text{ } \mu\text{m}^3. Multiplying the initial bacterium volume by this factor gives (2.0×10−12 cm3)×(1012 μm3/cm3)=2.0×100 μm3(2.0 \times 10^{-12} \text{ cm}^3) \times (10^{12} \text{ } \mu\text{m}^3/\text{cm}^3) = 2.0 \times 10^0 \text{ } \mu\text{m}^3.

Step-by-Step Solution

1
Determine the linear conversion factor between centimeters (cm\text{cm}) and micrometers (μm\mu\text{m}).
1 cm=10−2 m1 \text{ cm} = 10^{-2} \text{ m} and 1 μm=10−6 m1 \text{ } \mu\text{m} = 10^{-6} \text{ m}, so 1 cm=104 μm1 \text{ cm} = 10^4 \text{ } \mu\text{m}.
Before converting units of volume, the relationship between the linear units must be established.
2
Convert the linear conversion factor to a volume conversion factor by cubing it.
(1 cm)3=(104 μm)3=1012 μm3(1 \text{ cm})^3 = (10^4 \text{ } \mu\text{m})^3 = 10^{12} \text{ } \mu\text{m}^3.
Volume scales cubically with respect to linear dimensions.
3
Multiply the volume of the bacterium in cubic centimeters by the volume conversion factor.
(2.0×10−12 cm3)×(1012 μm3/cm3)=2.0×100 μm3(2.0 \times 10^{-12} \text{ cm}^3) \times (10^{12} \text{ } \mu\text{m}^3/\text{cm}^3) = 2.0 \times 10^0 \text{ } \mu\text{m}^3.
Multiplying by the conversion factor cancels the cubic centimeter units and yields the volume in cubic micrometers.

Key Concept

Volume Unit Conversion with Scientific Notation
Estimated Time:1m 30s
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Topics

Scientific Notation and Unit Conversions

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