Final answer:
The statement "The cost (C) of alcohol varies directly as its volume (v) in milliliters" is expressed as the equation C = kv, where C is the cost, v is the volume, and k is the cost per milliliter. This equation illustrates a direct variation, where the cost of alcohol increases proportionally with its volume.
Step-by-step explanation:
Direct Variation Equation
To express the statement "The cost (C) of alcohol varies directly as its volume (v) in milliliters" as a mathematical equation, we utilize the concept of direct variation. In direct variation, two variables change in the same manner such that when one variable increases, the other also increases proportionally. If we denote the constant of proportionality by k, then the equation can be written as:
C = kv
Here, C represents the cost of the alcohol, v represents the volume of the alcohol in milliliters, and k is a constant that represents the cost per milliliter of alcohol. This equation tells us that if you know the cost per milliliter (k), you can find the total cost (C) for any given volume (v) by multiplying the volume by the cost per milliliter.
For example, if the given concentration is 5% v/v alcohol, it means that there are 5 mL of alcohol dissolved in every 100 mL solution. To find the cost of a certain volume, you would simply multiply the volume (in milliliters) by the cost per milliliter. This direct variation equation is useful for predicting how changes in volume will affect the total cost.
In the context of other variables and relationships, it's important to identify the independent variable and the dependent variable. For instance, in a study determining whether weekly grocery bills change based on the number of family members, the number of family members would be the independent variable, and the weekly grocery bill would be the dependent variable.