91Ó°ÊÓ

In mathematics, a proportional relationship indicates that two values change at a consistent rate relative to each other. This means when one quantity changes, the other changes by a constant factor, known as the constant of proportionality or constant of variation. Using the example from the exercise, since \(y\) varies directly with \(t\) and the square of \(x\), it demonstrates a proportional relationship between these variables.

For a clearer understanding, here are a few points about proportional relationships:

• If one variable doubles, the other variable also doubles.
• This relationship is expressed using the equation form \(y = kx\), where \(k\) is the constant.
• The graph of a proportional relationship is always a straight line through the origin.

When two variables are proportionally related, understanding how changes affect them becomes much easier, allowing predictions about one based on the other.

Variation equations are mathematical expressions that describe how one quantity changes in relation to others. They can be direct, inverse, joint, or a combination. Direct variation is when one variable changes directly with another. In this case, as given in the exercise problem, \(y\) varies directly with both \(t\) and the square of \(x\).

The general form of a direct variation equation where a variable \(y\) is dependent on two or more factors is:

• \(y = k \, x_1 \, x_2 \ldots x_n\)

Here, \(k\) represents the constant of proportionality, and \(x_1, x_2, \ldots, x_n\) are independent variables. In our problem, the variation equation is not just limited to a direct relation but includes \(t\) and \(x^2\). Therefore,

Correct variation equation: \(y = k \, t \, x^2\).

This highlights how variation equations are employed to model the relationships between variables in real-world and mathematical scenarios.

Finding the constant \(k\), known as the constant of proportionality, is crucial for establishing an accurate variation equation. It helps in understanding how one variable is scaled by others in the relationship.

In the step-by-step solution provided, the constant \(k\) is derived by substituting known values into the variation equation. Here’s how:

• Start with the equation: \(y = k \, t \, x^2\).
• Insert the known values: \(y = 8\), \(t = 1\), and \(x = 4\).
• Substitute into the equation: \(8 = k \cdot 1 \cdot 4^2\).

From this equation, solving for \(k\) involves dividing both sides by 16 (since \(4^2 = 16\)). Therefore, \(k = \frac{8}{16} = \frac{1}{2}\).

Once \(k\) is determined, it is then used to construct the final equation: \(y = \frac{1}{2} \, t \, x^2\). This step is vital as it ensures that the equation accurately describes the relationship as provided by the problem's conditions.