Problem 14 - Algebra 1: Concepts and Skills

Question

Find the constant of variation. \(h\) varies directly with \(m,\) and \(h=112\) when \(m=12\)

Step-by-Step Solution

Answer

The constant of variation \(k\) is \(\frac{112}{12}\) or approximately 9.33.

Step 1: Understand the direct variation equation

In a direct variation, two variables are proportional. The formula for direct variation is \(y = kx\), where \(k\) is the constant of variation. Here, \(h\) varies directly with \(m\), so we can write the equation as \(h = km\).

Step 2: Substitute given values

We're given \(h=112\) when \(m=12\). Putting these values into the equation \(h = km\), we get 112 = \(k \cdot 12\).

Step 3: Solve for \(k\)

To find the value of \(k\), we divide both sides of the equation by 12. This gets us to the answer \(k= \frac{112}{12}\).

Key Concepts

Constant of Variation

The concept of the constant of variation is central to understanding direct relationships between two variables. In a direct variation, one variable is a constant multiple of the other. This constant multiplier is what we call the "constant of variation." When we have a formula like \(y = kx\), \(k\) represents the constant of variation. This simply means that no matter how \(x\), the independent variable, changes, \(y\) will always be \(k\) times that value. To find this constant, we substitute the known values from our exercise into the equation. For example, if \(h = 112\) when \(m = 12\), we can substitute these into the equation \(h = km\) to solve for \(k\). From here, finding \(k\) means we determine something about the nature of the relationship between \(h\) and \(m\). In this case, it turns out \(k\) is \(\frac{112}{12}\). This value tells us how much \(h\) increases for each unit of \(m\).

Proportional Relationship

A proportional relationship is a relationship between two variables where their ratios are constant. This occurs in direct variation scenarios, where \(y\) can be expressed as \(y = kx\). This equation shows us that \(y\) and \(x\) are in direct proportion to each other. In other words, when you divide \(y\) by \(x\), you get \(k\), the constant of variation. This constant ratio means that as one variable increases or decreases, the other does the same proportionally.

• If \(x\) doubles, \(y\) will double.
• If \(x\) is halved, \(y\) will be halved as well.

This kind of relationship is predictable and straightforward because the ratio remains the same regardless of how the values change. Recognizing and applying this proportional relationship, especially in direct variation problems, can simplify understanding how the two variables interact.

Solving Equations

In math, solving equations involves finding unknown values that satisfy the equation. Specifically, in direct variation, solving equations often centers around finding the constant of variation or unknown variables. In our example, we start with the equation \(h = km\). We know \(h = 112\) and \(m = 12\). To solve for \(k\), we substitute these values into the equation: \[112 = k \cdot 12\]. The next step is to isolate \(k\) by dividing both sides by 12:\[k = \frac{112}{12}\].By performing simple arithmetic, we find the value of \(k\). Solving such equations involves identifying known values and manipulating the equation to solve for any unknowns. Here, knowledge of basic algebra operations becomes incredibly useful to navigate and solve equations in direct variation scenarios.