Expert: Josh - 12/16/2008

Question
Josh I am pulling my hair out can you help me work out this problem....

Suppose a full 50 gallon drum contains a mixture of water and alcohol.   
a)  If the variable r denotes the density (weight/volume) of water and the density of the alcohol is 81% that of water write an expression for the density of the alcohol.  
b)  The total volume of the drum is the sum of the volumes of its contents. If the variable V denotes the volume of water in the drum then write an expression for the other part of the volume of the drum that is occupied by alcohol.  
c)  For any homogeneous material: weight is density times volume.  In terms of the variables defined above and the expressions you gave above, write an expression for the weight of water in the drum.  
d)  For any homogeneous material: weight is density times volume. Using the expressions for the density and volume of alcohol you gave above write an expression for the weight of alcohol in the drum.  
e) The total weight of the drum is the sum of the weights of its contents.  Write an expression for the total weight of the drum using the weights for the water and alcohol it contains that you gave above.  

Answer
Sonya

To ensure that you attempt these yourself, I will answer each of your questions indirectly.

a) It is standard practice to express the density of a substance relative to water. If it is denser than water, it will sink when thrown into a tank of water. In this case, its density is greater than 100% relative to water. If a substance has a density of 90% relative to water, it will float on water. Writing "r" for the density of water, this substance has a density of "0.9 r". Using this as an example, write an expression for the density of the alcohol, and call this "Ra".

b) There are 3 figures to look at.
(1) Volume of tank (is known to be) 50 gallon.
(2) Volume of water is represented by V.
(3) Volume of alcohol is the difference between (1) and (2).
Let the volume of alcohol be "Va". Va=50-V.

c) Recall (from part a) that the density of water is "r" and the volume of water (from part b) is "V". In terms of the volume of alcohol (the variable Va introduced in part b), the volume of water is given by V=50-Va ...let's call this (Eq.1). Now, for a homogeneous material, the density is uniform throughout. Writing "W" for the weight of water, W=r*V. Your remaining task is to substitute (Eq.1) for V in this equation, so that only Va and r appear on the right hand side. You must define/explain each symbol properly.

d) This is exactly a mirror situation to part c. Let "Wa" be the weight of the alcohol. Using [weight=density*volume], Wa=Ra*Va. Here, Ra represents the density of the alcohol you worked out in part a.

e) Weight in drum = W + Wa, it should follow from earlier results.