Problem 27 - Algebra 2 and Trigonometry

Question

In \(23-28,\) write an equation of the direct variation described. Water is flowing into a swimming pool at the rate of 25 gallons per minute. The number of gallons of water in the pool, \(g,\) is directly proportional to the number of minutes, \(m,\) that the pool has been filling from when it was empty.

Step-by-Step Solution

Answer

The equation of the direct variation is: \(g = 25m\).

Step 1: Identify Key Variables

In a direct variation problem, one quantity changes at a constant rate relative to another. Here, the number of gallons of water, \(g\), is directly proportional to the number of minutes, \(m\). This can be expressed as \(g = km\), where \(k\) is the constant of variation.

Step 2: Define the Constant of Variation

Since water is flowing into the pool at a rate of 25 gallons per minute, this rate is the constant of variation. Hence, \(k = 25\).

Step 3: Write the Direct Variation Equation

Substitute \(k = 25\) into the direct variation formula \(g = km\). Therefore, the equation becomes \(g = 25m\).

Key Concepts

Constant of Variation

When dealing with direct variation, the constant of variation is a crucial concept. It represents the fixed rate at which one variable changes in relation to another. In our example, water is flowing into a swimming pool at a steady rate. This means the number of gallons of water depends consistently on the time the pool is being filled. The mathematical representation involves a constant, often denoted by the letter "\(k\)." In this case, the function \(g = km\) is used, where \(g\) is the gallons of water and \(m\) is the time in minutes.

• "\(k\)" dictates the rate of change. If \(k\) is 25, then 25 gallons are added per minute.
• The greater the constant, the faster the water fills the pool.
• The unit of \(k\) depends on the variables involved (e.g., gallons per minute).

Understanding the constant of variation helps in comprehending how changes occur over time and how they are mathematically expressed.

Proportional Relationships

Proportional relationships are at the heart of direct variation problems. A relationship is considered proportional when two quantities always maintain a constant ratio. In simpler terms, whenever one quantity changes, the other changes in a predictable way, maintaining the same proportion. In our swimming pool scenario:

• The number of gallons \(g\) varies directly with the number of minutes \(m\).
• This relationship means for each increment of time, an equal amount of additional water is added.
• If the pool fills at 25 gallons per minute, then after 2 minutes, there would be 50 gallons, after 3 minutes, 75 gallons, and so on.

Proportional relationships allow for predictions: knowing one part of the proportion lets you calculate the other. This predictability is what makes direct variation equations powerful tools in real-world scenarios.

Linear Equations

Linear equations form the foundation of understanding how variables interact in a direct variation. In essence, these are mathematical representations of scenarios where the change between variables is consistent. Our equation, \(g = 25m\), is a classic example of a linear equation, where:

• "\(g\)" and "\(m\)" are the variables representing gallons and minutes, respectively.
• The equation is linear because its graph is a straight line.
• This line passes through the origin, emphasizing the direct proportionality.

In a direct variation expressed as a linear equation, if you plot the variables on a graph, you will see a straight line through the origin. This visualizes how each increment in one variable results in a consistent change in the other, affirming their linear relationship.