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3.3: Units Raised to a Power

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    365757

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    Learning Objectives
    • Given a quantity, convert from one set of units to another using dimensional analysis showing canceling on units. This includes numbers and units raised to a power.

    Conversion factors for area and volume can also be produced by the dimensional analysis method. Just remember that if a quantity is raised to a power of 10, both the number and the unit must be raised to the same power of 10. For example, to convert \(1500 \: \text{cm}^2\) to \(\text{m}^2\), we need to start with the relationship between centimeter and meter. We know that 1 cm = 10-2 m or 100 cm =1 m, but since we are given the quantity in 1500 cm2, then we have to use the relationship:

    \[1\, cm^2 = (10^{-2}\, m)^2 = 10^{-4}\, m^2\]

    CONCEPT MAP

    To convert centimeters squared to meters squared, use the conversion factor 0.01 meters per 1 centimeter, squared overall

    CALCULATION

    \[1500 \: \cancel{\text{cm}}^2 \times \left( \frac{10^{-2} \: \text{m}}{1 \: \cancel{\text{cm}}} \right)^2 = 0.15 \: \text{m}^2\]

    or

    \[1500 \: \cancel{\text{cm}}^2 \times \left( \frac{1 \: \text{m}}{100 \: \cancel{\text{cm}}} \right)^2 = 0.15 \: \text{m}^2\]

    or

    \[1500 \: \cancel{\text{cm}}^2 \times \frac{1 \: \text{m}^2}{10,000 \: \cancel{\text{cm}^2}} = 0.15 \: \text{m}^2\]

    Example \(\PageIndex{1}\): Unit Conversion

    The volume of a sphere is 333 in3. What is the volume in cubic cm (cm3)?

    Solution

    Steps for Problem Solving

    		 What is the volume of a sphere (radius 4.30 inches) in cubic cm (cm3)?
    Identify the "given” information and what the problem is asking you to "find."

    Given: 333 in3

    Find: cm3
    Determine other known quantities.

    1 in3 = (2.54 cm)3

    1 in3 = 16.4 cm3

    Prepare a concept map.

    		 To convert inches cubed to centimeters cubed, use conversion factor 2.54 centimeters per 1 inch, cubed overall

    Calculate.

    		 \(333 \cancel{in^3} \left(\frac{2.54cm}{1 \cancel{in}}\right)^3 = 5.46 \times10^3 cm^3\)

    Think about your result.

    		 A centimeter is a smaller unit than an inch, so the answer in cubic centimeters is larger than the given value in cubic inches.
    Exercise \(\PageIndex{1}\)

    Lake Tahoe has a surface area of 191 square miles. What is the area in square km (km2)?

    Answer
    495 km2

    Contributions & Attributions

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