Problem 15 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(V\) varies jointly as \(B\) and \(h\), and \(V=96\) when \(B=24\) and \(h=12\).

Step-by-Step Solution

Answer

The constant of variation is \(\frac{1}{3}\).

Step 1: Understand the Variation Relationship

The problem states that \(V\) varies jointly as \(B\) and \(h\). This means that \(V\) is directly proportional to the product of \(B\) and \(h\). The mathematical form of this relationship is given by:\[ V = kBh \]where \(k\) is the constant of variation we need to find.

Step 2: Substitute Known Values

We know that \(V = 96\), \(B = 24\), and \(h = 12\). Substitute these values into the variation equation:\[ 96 = k \cdot 24 \cdot 12 \]

Step 3: Solve for the Constant of Variation, \(k\)

To find \(k\), divide both sides of the equation \(96 = k \cdot 24 \cdot 12\) by \(24 \cdot 12\).Calculate \(24 \cdot 12 = 288\).Thus, the equation becomes:\[ k = \frac{96}{288} \]

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{96}{288}\) by dividing the numerator and the denominator by their greatest common divisor, which is 96.\[ \frac{96}{288} = \frac{1}{3} \]Therefore, \(k = \frac{1}{3}\).

Key Concepts

Constant of Variation

The constant of variation is a pivotal concept in understanding relationships between variables in contexts of proportionality. It basically quantifies the consistent factor that relates variables under a proportional statement. In our specific exercise, where volume (\(V\)) varies jointly with the base area (\(B\)) and height (\(h\)), the formula representing this is:

\[V = kBh\]Here, \(k\) stands as the constant of variation that connects the product of \(B\) and \(h\) to \(V\). Once you know \(k\), you have a complete picture of how changes in \(B\), and \(h\) affect \(V\).

The constant \(k\) is calculated using known values from a specific condition. This makes the constant of variation a great tool to predict or understand systems in mathematical modeling or physics, where specific relationships are maintained. It's like finding the glue that holds specified conditions together.

Direct Proportionality

Direct proportionality is a foundational concept where two variables increase or decrease together at the same rate. If one variable doubles, for example, the other doubles as well. This straightforward relationship ensures that the ratio between them remains constant.

In a joint variation scenario like the one in our exercise, direction proportionality extends to more than one variable. Specifically, we see that volume (\(V\)) varies jointly with both the base (\(B\)) and height (\(h\)):

• If either \(B\) or \(h\) increases while the other stays constant, \(V\) increases as well.
• Similarly, a decrease in either \(B\) or \(h\) leads to a decrease in \(V\).

This relationship is captured by the formula \(V = kBh\). The constancy in their proportional rates is ensured by the constant of variation \(k\). This principle of proportionality helps solve many real-world problems by modeling how one set of changes affects another.

Algebraic Equations

Algebraic equations serve as a bridge to express mathematical relationships in a structured and symbolic way. In the given exercise, the equation \(V = kBh\) represents the joint variation and reveals a multi-variable relationship.

Let's break down this equation:

• \(V\) is dependent on both \(B\) and \(h\).
• \(k\) acts as a modulator that adjusts \(V\) according to specific changes in \(B\) and \(h\).

By substituting specific values of \(V\), \(B\), and \(h\), we're able to solve for \(k\) and make predictions based on the equation.

Algebraic equations like this allow students not just to calculate, but to visualize the intrinsic relationships between variables. Through manipulation of these equations, complex relationships become clearer and our understanding of how elements interact strengthens. Algebra thus isn't just a tool for calculation, but a language to describe the world around us.