Question

a speck of dust is 5.0×10⁻⁷ m wide. which of the following is the best way to rewrite this quantity, using more appropriate units?
a. 5.0×10⁻¹⁰ km
b. 5.0×10⁻⁴ mm
c. 5.0×10⁻⁷ m
d. 5.0×10⁻⁵ cm

Explanation:

Step1: Recall length - unit conversion factors

We know that $1\ m = 100\ cm=1000\ mm = 10^{- 3}\ km$.

Step2: Convert $5.0\times10^{-7}\ m$ to centimeters

Using the conversion factor $1\ m = 100\ cm=10^{2}\ cm$, we multiply the given value by the conversion factor. So, $(5.0\times10^{-7}\ m)\times(10^{2}\ cm/m)=5.0\times10^{-7 + 2}\ cm=5.0\times10^{-5}\ cm$.

Step3: Convert $5.0\times10^{-7}\ m$ to millimeters

Using the conversion factor $1\ m=1000\ mm = 10^{3}\ mm$, we have $(5.0\times10^{-7}\ m)\times(10^{3}\ mm/m)=5.0\times10^{-7+3}\ mm = 5.0\times10^{-4}\ mm$.

Step4: Convert $5.0\times10^{-7}\ m$ to kilometers

Using the conversion factor $1\ m = 10^{-3}\ km$, we get $(5.0\times10^{-7}\ m)\times(10^{-3}\ km/m)=5.0\times10^{-7-3}\ km=5.0\times10^{-10}\ km$.
For a speck - of dust, millimeters or centimeters are more appropriate units than kilometers. Among the more appropriate units, $5.0\times10^{-4}\ mm$ is a more common way to express the size of a small particle like a speck of dust.

Answer:

B. $5.0\times10^{-4}\ mm$