Question

The density of water is $ 1.0\dfrac{g}{{mL}} $ . What is the density of water in $ \dfrac{{pounds}}{{gallon}} $ ?

Answer 1

Hint: The question of dimensional analysis. You have to convert $ \dfrac{g}{{mL}} $ into $ \dfrac{{pounds}}{{gallon}} $ that is $ \dfrac{{1g}}{{mL}} \times \mathop {\dfrac{?}{?}}\limits_{} = \dfrac{{pounds}}{{gal}} $ . The question mark is the conversion factor that you need to find out to convert. Remember that $ 1{\text{ }}ponds = 453.6g $ and $ 1{\text{ }}gallon = 3785.411{\text{ }}mL $ . Use these values to convert grams per milliliters to pounds per gallons.

Complete Step By Step Answer:
This is the question of dimensional analysis. The study of the relationship between physical quantities with the help of dimensions and units of measurement is termed dimensional analysis. The basic concept of dimension is that we can add and subtract only those quantities that have the same dimensions.
You need to convert $ \dfrac{g}{{mL}} $ into $ \dfrac{{pounds}}{{gallon}} $ . Find a conversion factor that can be multiplied with the $ \dfrac{g}{{mL}} $ units to give the value in $ \dfrac{{pounds}}{{gallon}} $ .
Here mass conversation unit $ g \to pounds $ and volume conversation unit $ mL \to gal $ is required. It is known that for mass, $ 1{\text{ }}ponds = 453.6g $ and for volume $ 1{\text{ }}gallon = 3785.411{\text{ }}mL $ . For our solving purpose we will write it as: $ \dfrac{{1pounds}}{{453.6g}} $ and $ \dfrac{{3785.411{\text{ }}mL}}{{1gal}} $ . These two will be our conversion factors that will be multiplied with the $ \dfrac{g}{{mL}} $ unit.
Now that we know the conversion factors to be multiplied, we will apply them and find out the final result.
 $ \dfrac{{1g}}{{mL}} \times \dfrac{{1\;pounds}}{{453.6g}} \times \dfrac{{3785.411\;mL}}{{1gal}} = \dfrac{{8.34pounds}}{{gal}} $
Therefore the density of water in $ \dfrac{{pounds}}{{gallon}} $ is $ \dfrac{{8.34pounds}}{{gal}} $ .

Note:
Dimensional analysis has limitations also. It doesn’t give information about the dimensional constant. It gives no information about whether a physical quantity is a scalar or vector. The formula containing trigonometric function, exponential functions, logarithmic function, etc. cannot be derived.