Problem 51 - Algebra 2

Question

Suppose that \(d\) varies jointly with \(r\) and \(t,\) and \(d=110\) when \(r=55\) and \(t=2\) . Find \(r\) when \(d=40\) and \(t=3 .\)

Step-by-Step Solution

Answer

The value of \(r\) when \(d=40\) and \(t=3\) is approximately 13.33

Step 1: Find the constant of variation \(k\)

We can use the first given condition \(d=110\), \(r=55\) and \(t=2\) to find the constant \(k\) from the equation \(d = k \cdot r \cdot t\). So solving for \(k\), we get \[ k = \frac{d}{r \cdot t} = \frac{110}{55 \cdot 2} = 1 \]

Step 2: Find \(r\) when \(d=40\) and \(t=3\)

Now use the given condition \(d=40\) and \(t=3\) along with the constant of variation \(k=1\) that we found in step 1 to find \(r\). This can be done by rearranging equation \(d = k \cdot r \cdot t\) for \(r\) and substituting the known values into the equation, yielding \[ r = \frac{d}{k \cdot t} = \frac{40}{1 \cdot 3} = \frac{40}{3} \approx 13.33 \]

Key Concepts

Constant of Variation

The concept of joint variation introduces the idea of a constant called the **constant of variation**. When we say that a variable "varies jointly" with other variables, it means that this variable is proportional to the product of these other variables. The equation for joint variation is given as\[ d = k \, r \, t \]where:

• \(d\) is the dependent variable,
• \(r\) and \(t\) are the independent variables, and
• \(k\) is the constant of variation.

The constant \(k\) is an important value because it essentially scales the relationship between the variables. It means, for example, if \(d\) increases, either an increase in \(r\), \(t\), or \(k\), or a combination of those will be responsible. Finding the constant of variation \(k\) is crucial when solving problems involving variation, as it allows us to express the dependent variable in terms of the independent variables.

Solving for Variables

In problems involving joint variation, once the constant of variation \(k\) is determined, other variables can be discovered through algebraic manipulation of the original equation. Given the joint variation formula\[ d = k \, r \, t \]if we need to solve for one of the independent variables, say \(r\), we can rearrange the equation to isolate \(r\) as follows:\[ r = \frac{d}{k \, t} \]With the known values of \(d\), \(k\), and \(t\), it's straightforward to substitute and solve for \(r\). This practical problem-solving approach, using substitution and rearrangement, is central in algebra and allows complex relationships to be simplified and understood. Additionally, it underscores the flexibility of algebraic equations, where solving for different variables requires simply reconfiguring the given formula.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between expressions and are foundational to solving joint variation problems. The essential skill in working with algebraic equations is understanding how variables relate to one another and how equations can be manipulated to isolate a specific variable.When dealing with joint variation scenarios, your equation becomes a tool to navigate through the relationships among all involved variables. Consider the equation from joint variation:\[ d = k \, r \, t \]It represents a powerful connection among \(d\), \(k\), \(r\), and \(t\). Algebra enables us to rearrange this equation to uncover the values of individual variables. For instance, provided we know three out of four variables, we can reorganize this equation into different forms to solve for the missing one:

• \(k = \frac{d}{r \, t}\)
• \(r = \frac{d}{k \, t}\)
• \(t = \frac{d}{k \, r}\)

These manipulated equations serve as stepping stones to solutions, facilitating a deeper understanding of the problem's structure and aiding in the systematic solving of similar algebraic challenges.