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When we say a variable varies directly with another variable, it means that as one variable increases, the other variable increases at a constant rate. In mathematical terms, if variable \(r\) varies directly with variable \(m^2\), we can write this relationship as:

• \[ r = k m^2 \]

here, \(k\) is the proportionality constant. It indicates the rate at which \(r\) changes with respect to \(m^2\). In the given problem, the square of \(m\) directly affects the value of \(r\). For example, if you increase \(m\), \(m^2\) increases, which in turn increases \(r\).

Inverse variation means that as one variable increases, the other variable decreases proportionally. For the given problem, \(r\) varies inversely with \(s\), which can be written as:

• \[ r = k \frac{1}{s} \]

here, \(k\) is again the proportionality constant. This relationship tells us that if \(s\) increases, the value of \(r\) decreases, assuming \(k\) is constant. In our problem, \(r\) depends inversely on \(s\), which means higher values of \(s\) will make \(r\) smaller. Think of it as sharing a fixed number of resources among more people; each person gets less.

The proportionality constant \(k\) is a crucial part of both direct and inverse variations because it helps bridge the relationship between the variables.

Finding \(k\)

In the given problem, \(k\) was determined using the provided values: \(r = 12\), \(m = 6\), and \(s = 4\). We substitute these values into the combined variation formula:

• \[ r = k \frac{m^2}{s} \]

to solve for \(k\).

Using \(k\)

Once \(k\) is known, it can be used to find the value of \(r\) for any given \(m\) and \(s\). This is done by plugging the new values of \(m\) and \(s\) into the equation and solving for \(r\). This helps predict how changes in \(m\) or \(s\) will affect \(r\). In our problem, with \(k = \frac{4}{3}\), we could easily find \(r\) for new values of \(m = 6\) and \(s = 20\). Understanding this constant is key to solving variation problems accurately.