Question

write a general formula to describe each variation. The square of \(T\) varies directly with the cube of \(a\) and inversely with the square of \(d ; T=2\) when \(a=2\) and \(d=4\)

Short answer

The general formula is \( T^2 = 8 \frac{a^3}{d^2} \).

Step-by-step solution

Identify the type of variation

The problem states that the square of \( T \) varies directly with the cube of \( a \) and inversely with the square of \( d \). This indicates a combined variation.

Set up the initial variation formula

Based on the description, the formula can be written as: \[ T^2 = k \frac{a^3}{d^2} \] Where \( k \) is the constant of proportionality.

Substitute the given values to find \( k \)

We are given that \( T = 2 \), \( a = 2 \), and \( d = 4 \). Substitute these values into the variation formula: \[ 2^2 = k \frac{2^3}{4^2} \] Simplify the equation to solve for \( k \).

Solve for the constant \( k \)

Simplify the equation: \[ 4 = k \frac{8}{16} \] This simplifies to: \[ 4 = k \frac{1}{2} \] So, \( k = 4 \times 2 = 8 \).

Write the general formula

Substitute the value of \( k \) back into the initial variation formula: \[ T^2 = 8 \frac{a^3}{d^2} \] This is the general formula describing the variation.

Key concepts

Direct Variation

Direct variation occurs when one variable increases, the other also increases in proportion. It forms a simple linear relationship. This relationship can be expressed with the equation: \[ y = kx \]Here, \( k \) is known as the proportionality constant. All changes in one variable directly affect another. For example, if \( y \) varies directly as \( x \), doubling \( x \) would also double \( y \). Practical situations include speed and distance in uniform motion, or cost of products and their quantity.

Inverse Variation

Inverse variation is the opposite of direct variation. Here, as one variable increases, the other decreases proportionally. The relationship is given by: \[ y = \frac{k}{x} \] As before, \( k \) is the proportionality constant. This means multiplying \( x \) by a factor reduces \( y \) by the same factor. An example is the relationship between speed and travel time for a constant distance. If you increase your speed, the travel time decreases accordingly. In everyday life, this could be observed in situations like the strength of internet signal decreasing with increased distance from the router.

Proportionality Constant

The proportionality constant \( k \) is a key part of both direct and inverse variations. It defines how strongly variables are related.In direct variation, \( k \) shows the rate of increase. The larger \( k \), the steeper the increase in \( y \) with \( x \). In inverse variation, \( k \) determines how quickly \( y \) drops as \( x \) rises. To find \( k \), substitute known values for the variables in the variation equation. For example, if you know \( y \) and \( x \) for direct variation: \[ k = \frac{y}{x} \]. Accurately finding \( k \) ensures your variation equation models real-world situations precisely.

Algebraic Formulas

In algebra, formulas describe how variables relate. Direct and inverse variations are fundamental examples of these relationships. The combined variation is a good practical example, where variables vary both directly and inversely. The formula from the exercise, \[ T^2 = k \frac{a^3}{d^2} \], shows this. Here:

• \(T^2\) is directly proportional to \(a^3\), meaning as \(a\) increases, \(T\) also increases.
• \(T^2\) is inversely proportional to \(d^2\), so as \(d\) increases, \(T\) decreases.

In this formula, \( k \) once again is the proportionality constant, calculated as 8 from the given values, making the relationship concrete and simplified.