## Q. 2.13

Converting Quantities Involving Units Raised to a Power

A circle has an area of 2659 $cm^{2}$. What is its area in square meters?

## Verified Solution

 SORT You are given an area in square centimeters and asked to convert the area to square meters. GIVEN: 2659 $cm^{2}$ FIND: $m^{2}$ STRATEGIZE Build a solution map beginning with $cm^{2}$ and ending with $m^{2}$. Remember that you must square the conversion factor. SOLUTION MAP $cm^{2}\underset{\frac{(0.01 m)^2}{(1 cm)^2} }{\longrightarrow } m^{2}$ RELATIONSHIPS USED 1 cm = 0.01 m (from Table 2.2) SOLVE Follow the solution map to solve the problem. Square the conversion factor (both the units and the number) as you carry out the calculation. Round the answer to four significant figures to reflect the four significant figures in the given quantity. The conversion factor is exact and therefore does not limit the number of significant figures. SOLUTION $2659 cm^2\times \frac{(0.01 m)^2}{(1 cm)^2} =2659 \cancel{cm^2}\times \frac{10^{-4} m^2}{1 \cancel{cm^2}}$   = 0.265900 $m^{2}$    = 0.2659 $m^{2}$ CHECK Check your answer. Are the units correct? Does the answer make physical sense? The units of the answer are correct, and the magnitude makes physical sense. A square meter is much larger than a square centimeter, so the value in square meters should be much smaller than the value in square centimeters.

Table 2.2

 SI Prefix Multipliers Prefix Symbol Meaning Multiplier tera- T trillion 1,000,000,000,000 $(10^{12})$ giga- G billion 1,000,000,000 $(10^{9})$ mega- M million 1,000,000 $(10^{6})$ kilo- k thousand 1,000 $(10^{3})$ hecto- h hundred 100 $10^{2}$ deca- da ten 10 $10^{1}$ deci- d tenth 0.1 $(10^{-1})$ centi- c hundredth 0.01 $(10^{-2})$ milli- m thousandth 0.001 $(10^{-3})$ micro- µ millionth 0.000001 $(10^{-6})$ nano- n billionth 0.000000001 $(10^{-9})$ pico- p trillionth 0.000000000001 $(10^{-12})$ femto- f quadrillionth 0.000000000000001 $(10^{-15})$