Problem 12 - College Algebra

Question

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(V\) varies jointly as \(h\) and the square of \(r\)

Step-by-Step Solution

Answer

The joint variation equation is \(V = khr^2\), where \(k\) is the constant of variation.

Step 1: Understanding Joint Variation

If a variable \(V\) varies jointly as \(h\) and \(r^2\), then \(V\) is directly proportional to the product of \(h\) and \(r^2\). This is represented as \(V = khr^2\), where \(k\) is the constant of variation.

Step 2: Writing the Equation

Given that \(V\) varies jointly as \(h\) and the square of \(r\), the equation that represents this relationship is \(V = khr^2\). Here, \(k\) can be any non-zero constant.

Key Concepts

Direct Proportionality

Direct proportionality is a fundamental concept in mathematics that describes a relationship between two variables where one is a constant multiple of the other. In such cases, when one variable increases, the other increases at a fixed rate, and if one decreases, the other does so at the same rate. This is visually represented by a straight line passing through the origin on a graph.

For instance, if we say that variable \( V \) is directly proportional to the product of \( h \) and \( r^2 \), we mean \( V \) changes at a consistent rate depending on the combined value of \( h \) and \( r^2 \). This relationship can be mathematically expressed as:

• \( V = k imes (h imes r^2) \)

here, \( k \) represents the constant of proportionality or the constant of variation, providing the fixed rate of change.

Constant of Variation

The constant of variation, often denoted by \( k \), plays a crucial role in equations of direct proportionality. It serves as the scaling factor that determines how the variables relate to one another.

In the context of joint variation, like in our exercise, the constant \( k \) determines how much \( V \) will change in response to changes in \( h \) and \( r^2 \). For every increase in \( h \) or \( r^2 \), \( V \) will increase by a factor of \( k \). This means:

• When \( k \) is larger, \( V \) changes more aggressively with changes in \( h \) and \( r^2 \).
• Conversely, a smaller \( k \) results in \( V \) changing more gently.

The constant can be any non-zero number, and it is a crucial part of the equation \( V = khr^2 \) since it maintains the proportional relationship between the variables.

Square of a Variable

When we talk about the square of a variable, we're discussing raising that variable to the power of two. This means multiplying the variable by itself, like \( r^2 = r \times r \). Squaring a variable is a common operation in many mathematical equations, especially when describing geometric areas and joint variability problems like this one.

In the equation \( V = khr^2 \), the square of \( r \) signifies that small changes in \( r \) can lead to larger changes in \( V \). This happens because the effect of \( r \) on \( V \) is not linear; instead, it is compounded because of the squaring.

• As \( r \) increases, \( r^2 \) grows quadratically, leading to a faster increase in \( V \).
• Similarly, decreases in \( r \) more drastically decrease \( r^2 \), resulting in a quicker reduction in \( V \).

Understanding the square of a variable helps in comprehending the dynamic changes in joint variation equations like these.