Problem 17 - Beginning and Intermediate Algebra

Question

Write a general variation equation using \(k\) as the constant of variation. \(Q\) varies directly as the square root of \(z\) and inversely as \(m\)

Step-by-Step Solution

Answer

The general variation equation for the given relationship is \( Q = k \frac{\sqrt{z}}{m} \), where \(Q\) varies directly with the square root of \(z\) and inversely with \(m\), and \(k\) is the constant of variation.

Step 1: Determine the type of variation

Since \(Q\) varies directly with \(\sqrt{z}\) and inversely with \(m\), we are dealing with a combined variation. The general form of such an equation is \(Q = k\frac{A}{B}\), where \(Q\), \(A\), and \(B\) are the variables, and \(k\) is the constant of variation.

Step 2: Express the relationship between the variables

We are given that \(Q\) varies directly with \(\sqrt{z}\) and inversely with \(m\). This means that as \(\sqrt{z}\) increases, \(Q\) increases, while an increase in \(m\) causes a decrease in \(Q\). Using the general form of the combined variation equation, we express the relationship as: \[ Q = k \frac{\sqrt{z}}{m} \]

Step 3: Write the final general variation equation

Now that we have expressed the relationship between the variables \(Q\), \(z\), and \(m\), we have the final general variation equation as: \[ Q = k \frac{\sqrt{z}}{m} \] This equation represents the combined variation between \(Q\), \(\sqrt{z}\), and \(m\) with \(k\) as the constant of variation.

Key Concepts

Direct Variation

Direct variation describes a scenario where two variables are related in such a way that when one variable increases, the other also increases, and vice versa. This relationship can be expressed mathematically through a formula. The formula often takes the form of \( y = kx \). Here:\

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• \(y\) is the dependent variable.
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• \(x\) is the independent variable.
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• \(k\) is the constant of variation.
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\In the exercise, we have direct variation with respect to \( \sqrt{z} \). This means that \(Q\) increases as \(\sqrt{z}\) increases, assuming other factors remain constant. The presence of \( \sqrt{z} \) in the numerator of the equation \( Q = k \frac{\sqrt{z}}{m} \) aligns with this pattern of direct variation.

Inverse Variation

Inverse variation occurs when one variable increases while the other variable decreases. This type of relationship can be expressed by the formula \( y = \frac{k}{x} \). Here:\

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• \(y\) is the dependent variable.
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• \(x\) is the independent variable.
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• \(k\) is the constant of variation.
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\In the given exercise, \(Q\) varies inversely with \(m\). This means that as \(m\) increases, \(Q\) decreases. The presence of \(m\) in the denominator of the equation \( Q = k \frac{\sqrt{z}}{m} \) translates this relationship in mathematical form.

Constant of Variation

The constant of variation, represented by \(k\), plays a central role in both direct and inverse variations. It defines how large the correlation between the variables is. The constant remains unchanged as the values of variables shift. It acts as a proportionality factor in the corresponding equations.\
\For example, in the general equation of the exercise \( Q = k \frac{\sqrt{z}}{m} \), \(k\) dictates the extent to which \(Q\) changes concerning variations in \(\sqrt{z}\) and \(m\). Understanding this constant enables you to predict changes in variable relationships accurately.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. In this exercise, an algebraic equation is used to express the direct and inverse relationships of the variables.\
The formula \( Q = k \frac{\sqrt{z}}{m} \) represents combined variation, where the variable \(Q\) is determined through other operations (square rooting and division) with \(z\) and \(m\).\

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• "Direct variation" is portrayed by the presence of \(\sqrt{z}\) alongside the constant \(k\).
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• "Inverse variation" is represented by \(m\) in the denominator.
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\Solving such equations often involves substituting known variable values and solving for unknowns. These equations are foundational in understanding relationships in various scientific fields.