⚙️ Learning Objectives To convert a value reported in one unit raised to a power of 10, to a corresponding value in a different unit raised to the same power of 10, using conversion factors. Conversion factors for area and volume may also be produced using dimensional analysis. 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 cm 2 to 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 cm 2 , then we have to use the relationship: 1 cm 2 = (10 –2 m) 2 = 10 –4 m 2 CONCEPT MAP \({\color[rgb]{0.8, 0.0, 0.0}\boxed{\;\;\mathrm{cm}^2\;\;}}\;\left(\xrightarrow[{1\;\mathrm{cm}}]{10^{-2}\;\mathrm m}\right)^2\;{\color[rgb]{0.0, 0.0, 1.0}\boxed{\;\;\;\mathrm m^2\;\;\;}}\) CALCULATION \(1500\:\cancel{\text{cm}^2}\times\left(\dfrac{10^{-2}\:\text{m}}{1\:\cancel{\text{cm}}}\right)^2=\boxed{0.15\:\text{m}^2}\) or \(1500\:\cancel{\text{cm}^2}\times\left(\dfrac{1\:\text{m}}{100\:\cancel{\text{cm}}}\right)^2=\boxed{0.15\:\text{m}^2}\) or \(1500\:\cancel{\text{cm}^2}\times\dfrac{1\:\text{m}^2}{10,000\:\cancel{\text{cm}^2}}=\boxed{0.15\:\text{m}^2}\) ✅ Example \(\PageIndex{1}\) What is the volume of a sphere (radius = 4.30 inches) in cubic centimeters (cm 3 ), given \(V=\frac43\mathrm\pi r^3\)? Solution Steps for Problem Solving Result Identify the "given” information and what the problem is asking you to "find." Given: radius = 4.30 in Find: cm3 (volume) List other known quantities. Metric prefixes are found in 2.5: The Metric System. Other conversions may be found in Appendix 2: Conversions. \(V=\frac43\mathrm\pi r^3\) = \(\frac43\times\;3.1416\;\times(4.3\underline0\;\mathrm{in})^3\) = \(33\underline3.04\;\mathrm{in}^3\) 1 in = 2.540 cm Prepare a concept map and use the proper conversion(s). \({\color[rgb]{0.8, 0.0, 0.0}\boxed{\;\;\;\mathrm{in}^3\;\;\;}}\;\left(\xrightarrow[{1\;\mathrm{in}}]{2.540\;\mathrm{cm}}\right)^3\;{\color[rgb]{0.0, 0.0, 1.0}\boxed{\;\;\;\mathrm{cm}^3\;\;\;}}\) Calculate the answer. \(33\underline3.04\;\cancel{\mathrm{in}^3}\;\times\;\left(\dfrac{2.540\;\mathrm{cm}}{1\;\cancel{\mathrm{in}}}\right)^3\;=\;\boxed{5.46\times10^3\;\mathrm{cm}^3}\) 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 kilometers (km 2 )? Answer 494 km 2 This page is shared under a CK-12 license and was authored, remixed, and/or curated by Melissa Alviar-Agnew, Henry Agnew, and Lance S. Lund (Anoka-Ramsey Community College). Original source: https://www.ck12.org/c/chemistry/ . LICENSED UNDER