Problem 29 - Algebra A Combined Function

Question

Write each statement as an equation. Use \(k\) as the constant of variation. \(r\) varies jointly as \(s\) and the cube of \(t\).

Step-by-Step Solution

Answer

The equation is \(r = k \cdot s \cdot t^3\).

Step 1: Identify Joint Variation

The problem states that one variable, \(r\), varies jointly as two other variables, \(s\) and the cube of \(t\). Joint variation refers to a situation where a variable depends on the product of two or more other variables.

Step 2: Use Variation Formula

For joint variation, the general formula is \(r = k \cdot s \cdot t^n\) where \(k\) is the constant of variation, and \(n\) is the power to which one of the variables (in this case, \(t\)) is raised.

Step 3: Substitute Values

Since \(r\) varies jointly as \(s\) and the cube of \(t\), we replace \(n\) with 3: \(r = k \cdot s \cdot t^3\). This equation represents the relationship between the three variables according to the given joint variation.

Key Concepts

constant of variation

In joint variation problems, understanding the role of the constant of variation, often represented by the symbol \(k\), is key. This constant is the factor that scales the relationship between the variables. Essentially, it tells us how strong the relationship is between these variables.
For example, in our joint variation problem, we have the equation \(r = k \cdot s \cdot t^3\). Here, \(k\) shows how \(r\) scales with respect to \(s\) and \(t^3\).

• If \(k\) is large, small changes in \(s\) or \(t^3\) will cause large changes in \(r\).
• Conversely, if \(k\) is small, \(r\) is less sensitive to changes in \(s\) and \(t^3\).

It acts almost like a 'tuning knob' for how the input variables affect the output variable.
Finding \(k\) usually involves having data points or additional conditions provided, which can allow us to solve for \(k\) as an unknown variable in the equation.

equation formulation

Formulating the correct equation in a joint variation scenario involves translating words into the mathematical language. The exercise requires us to express the relationship as a mathematical equation using the given variables.
Firstly, identify the type of variation. Joint variation, as described in the task, means a variable depends on the product of two or more other variables. In our case, \(r\) is directly related to \(s\) and \(t^3\).

• The equation structure typically looks like \(r = k \cdot s \cdot t^n\), where \(n\) indicates the power of \(t\).
• In the problem statement, \(t\) is specified to be cubed, thus \(n = 3\).

This results in the joint variation equation \(r = k \cdot s \cdot t^3\). Such formulation is crucial because it precisely encapsulates the dependency of \(r\) on the other variables and their respective powers.

variables in algebra

Variables are foundational components in algebra that represent numbers or values that can change. In the context of joint variation, they are crucial in establishing relationships between quantities.
The variables used in the exercise include \(r\), \(s\), and \(t\).

• \(r\) is the dependent variable, meaning its value is affected by changes in the other variables.
• \(s\) and \(t\) are independent variables, which directly influence \(r\), with \(t\) specifically having an exponential influence as \(t^3\).

In algebra, each variable symbolizes an unknown that we are often trying to solve for, and it could be any number or expression. They allow us to generalize and solve real-world problems mathematically. Through manipulating these variables in equations, we can observe and predict the behavior of quantities in a system.