Problem 15 - Beginning and Intermediate Algebra

Question

Write a general variation equation using \(k\) as the constant of variation. \(s\) varies jointly as \(r\) and \(t\)

Step-by-Step Solution

Answer

The general variation equation for the given problem is \(s = k(rt)\), where \(s\) varies jointly as \(r\) and \(t\) with a constant of variation \(k\).

Step 1: Understand the concept of joint variation

When a variable, in this case, s, varies jointly as r and t, it means that s is directly proportional to the product of r and t. In other words, as the product of r and t increases or decreases, s will also increase or decrease by the same proportion. This relationship can be represented by the formula: \[s = k(rt)\] where k is the constant of variation.

Step 2: Formulate the general equation

Now that we know the relationship between s, r, and t, we can write a general variation equation using k as the constant of variation. Using the formula from step 1: \[s = k(rt)\] Our general variation equation is: \[s = k(rt)\] This equation shows that s varies jointly as r and t with a constant of variation k.

Key Concepts

Direct Proportionality in Joint Variation

Direct proportionality is a fundamental concept in understanding joint variation. When we say a variable varies directly with another, it means that any change in one variable results in a proportional change in the other. In the context of joint variation, direct proportionality extends to more than two variables. Here it involves a situation where a variable, say \( s \), is directly proportional to the product of two other variables, \( r \) and \( t \).
You can imagine this as a group of variables working together in tandem. If \( r \) or \( t \) doubles, the product \( rt \) doubles, and consequently, \( s \) also doubles, assuming the constant of variation remains unchanged.
Joint variation can be visualized with expressions like \( s \propto rt \). This expression can be written as \( s = k(rt) \) to show how the variables are interconnected with the constant \( k \).

Understanding the Constant of Variation

The constant of variation, denoted by \( k \), is an important element in variation equations. It acts as a scaling factor that adjusts the magnitude of the relationship between the variables. The beauty of this constant is that it remains unchanged as long as the conditions of the scenario are constant.
In a joint variation equation like \( s = k(rt) \), \( k \) determines how much \( s \) will change for each unit change in the product \( rt \). It's crucial to understand that though \( s \) varies with \( rt \), the constant \( k \) will ensure that the "rate" of this variation is consistently applied.

• Consider \( k \) as a multiplier that modifies the direct proportional effects of \( rt \).
• A larger \( k \) value increases the rate at which \( s \) changes, whereas a smaller \( k \) means \( s \) changes less rapidly.

Finding \( k \) often involves considering specific values of \( s \), \( r \), and \( t \) to solve for it based on known conditions.

Formulating and Using Variation Equations

A variation equation is a powerful tool for expressing relationships in joint variation scenarios. It allows you to succinctly capture the essence of how variables interact. The general form of a joint variation equation is \( s = k(rt) \).
This equation signifies that \( s \) responds predictably to the product of \( r \), \( t \), and the constant of variation \( k \). In practical applications, knowing how to form and use these equations provides insight on how changes in one area impact another.

• For instance, if you find that \( s = 50 \) when \( r = 5 \) and \( t = 2 \), you can solve for \( k \) by rearranging the formula as \( k = \frac{50}{5 \times 2} \), which simplifies to \( k = 5 \).
• After determining \( k \), you can predict the value of \( s \) for different values of \( r \) and \( t \).

This process underscores the utility of variation equations in solving real-world problems by showcasing dependency relations cleanly and efficiently.