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 unit - conversion factors

We know that $1\ m = 100\ cm=1000\ mm$ and $1\ km = 1000\ m$.

Step2: Convert meters to centimeters

To convert from meters to centimeters, use the conversion factor $1\ m=100\ cm = 10^{2}\ cm$. So, if we have a length $L = 5.0\times10^{-7}\ m$, then $L=(5.0\times10^{-7})\times10^{2}\ cm=5.0\times10^{-5}\ cm$.

Step3: Convert meters to millimeters

To convert from meters to millimeters, use the conversion factor $1\ m = 1000\ mm=10^{3}\ mm$. So, $L=(5.0\times10^{-7})\times10^{3}\ mm = 5.0\times10^{-4}\ mm$.

Step4: Convert meters to kilometers

To convert from meters to kilometers, use the conversion factor $1\ km=1000\ m = 10^{3}\ m$. So, $L=(5.0\times10^{-7})\div10^{3}\ km=5.0\times10^{-10}\ km$.

Step5: Determine the most appropriate unit

For a speck of dust, millimeters are a more appropriate unit compared to kilometers or meters.

Answer:

B. $5.0\times 10^{-4}\ mm$