3.3: Units Raised to a Power
- Page ID
- 365757
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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
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
\[1500 \: \cancel{\text{cm}}^2 \times \frac{1 \: \text{m}^2}{10,000 \: \cancel{\text{cm}^2}} = 0.15 \: \text{m}^2\]
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. |
 |
|
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. |
Lake Tahoe has a surface area of 191 square miles. What is the area in square km (km2)?
- Answer
- 495 km2
Contributions & Attributions
This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:
Henry Agnew (UC Davis)