Problem 21 - Algebra and Trigonometry

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. M varies directly as \(x\) and inversely as \(y .\) If \(x=2\) and \(y=6\) , then \(M=5\)

Step-by-Step Solution

Answer

The constant of proportionality k is 15.

Step 1: Understand the Problem

We need to express the statement 'M varies directly as x and inversely as y' as an equation. This means M is directly proportional to x and inversely proportional to y.

Step 2: Express the Relationship as an Equation

Since M varies directly as x and inversely as y, we can write this as:\[ M = k \frac{x}{y} \]where \( k \) is the constant of proportionality.

Step 3: Substitute Given Values to find k

We are given that when \( x = 2 \), \( y = 6 \), and \( M = 5 \). Substitute these values into the equation:\[ 5 = k \frac{2}{6} \]which simplifies to:\[ 5 = k \frac{1}{3} \]

Step 4: Solve for Constant of Proportionality k

Solve the equation from Step 3 for \( k \):Multiply both sides by 3 to isolate \( k \):\[ 5 \times 3 = k \]\[ 15 = k \]Thus, the constant of proportionality \( k \) is 15.

Key Concepts

Proportionality Constant

In the world of mathematics, a proportionality constant is an important factor that connects two varying quantities. It serves as the bridge that defines how one variable changes concerning the other. Let's break it down further:

• When two quantities are directly linked together in some manner, they rise or fall in a synchronized fashion.
• The proportionality constant, usually denoted as \( k \), describes the strength and nature of this relationship.

Think of \( k \) as a scaling factor. In our example, where one quantity depends on both a direct and an inverse element, \( k \) becomes crucial. For instance, we have the equation:\[ M = k \frac{x}{y}\]Here, \( k \) helps us understand exactly how much \( M \) will grow when \( x \) increases or \( y \) decreases. In essence, once we've nailed down \( k \), predicting other future relationships within the same expression becomes much simpler.

Direct Variation

Direct Variation is a common relationship in mathematics which occurs when one quantity grows while the other also grows. It's quite intuitive – as if one is paired with the other like dance partners.

• If \( M \) increases with \( x \), then \( M \) is directly proportional to \( x \).
• This means in an equation like \( M = kx \), \( M \) is directly varying with \( x \) and \( k \) is positive.

The direct variation indicates a linear relationship. In our specific problem, \( M \) directly varies with \( x \), meaning if \( x \) were to double, and if \( y \) stays constant, \( M \) would also double, scaled by our proportionality constant \( k \). The key idea is this simplicity of movement in direct proportion:

• If one goes up, so does the other, maintaining a constant ratio.

Inverse Variation

Inverse Variation is where things get more intricate. In this form of variation, one quantity increases while the other decreases, similar to a seesaw balancing act.

• When \( M \) varies inversely as \( y \), if \( y \) increases, then \( M \) decreases.

This kind of relationship is expressed mathematically as:\[ M = \frac{k}{y}\]Think of it as two people pulling opposite sides of a rope. The stronger one pulls, the more the other moves in the opposite direction. In the equation setup in the original problem, \( M \) is inversely dependent on \( y \):

• If \( y \) becomes larger, \( M \) shrinks, as long as \( x \) remains constant.

It's crucial to recognize how this inverse relationship affects overall outcomes. Malbalance in one component will directly change the resulting position or value of the other within the defined proportionality, influenced by our constant \( k \).