Question

Write a general formula to describe each variation. \(V\) varies directly with \(x^{3} ; \quad V=36 \pi\) when \(x=3\)

Short answer

The general formula is \( V = \frac{4 \pi}{3} \times x^3 \).

Step-by-step solution

Understand Direct Variation

In direct variation, one variable changes directly as another variable changes. The general form is given by: \( V = k \times x^3 \) where \(k\) is the constant of variation to be determined.

Substitute Given Values

We are given that \( V = 36 \pi \, \ \) when \( x = 3\). Substitute these values into the general form to find \( k \): \( 36 \pi = k \times 3^3 \).

Solve for the Constant \(k\)

Simplify the equation to solve for \( k \): \( 36 \pi = k \times 27 \ \) \( k = \frac{36 \pi}{27} \ \) \( k = \frac{4 \pi}{3} \)

Write the General Formula

Substitute the value of \( k \) back into the general form of the direct variation equation: \( V = \frac{4 \pi}{3} \, \times x^3 \)

Key concepts

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Direct variation is a fundamental mathematical concept. In direct variation, one variable changes proportionally with another. When one variable increases, the other does as well, following a consistent ratio. The placeholder variable often used is \(k\), the constant of proportionality. The formula for direct variation can be written as: \( V = k \times x^3 \) when considering cube variations. Whenever values for \(V\) and \(x\) are given, you can substitute them to find the constant \(k\). This foundational knowledge helps you understand how different quantities relate directly to one another.

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Solving for constants in a direct variation formula is crucial. You must be given certain values for \(V\) and \(x\) to find the constant \(k\). For example, if \( V = 36\pi \) when \( x = 3 \), you substitute these into the formula \( 36\pi = k \times 3^3 \). By simplifying, you get \( k = \frac{36\pi}{27} \), which simplifies further to \( k = \frac{4\pi}{3} \). Finding the constant allows you to create a complete formula to predict values of \(V\) for any \(x\). This constant represents the rate of change between variables and is essential for solving practical problems.

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In algebra, general form equations represent the relationship between variables in a standard structure. For direct variation with cubes, the general form is \( V = k \times x^3 \). Once you determine the constant \(k\) from specific values, you substitute it back into the formula. For instance, if you discovered \( k = \frac{4\pi}{3} \) from the step before, the equation becomes \( V = \frac{4\pi}{3} \times x^3 \). This general form is valuable because it lets you easily compute \(V\) for any \(x\). Recognizing and using these forms helps predict outcomes systematically.

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Algebraic manipulation involves changing the form of an equation to find specific values, like constants. For direct variation, you often need to rearrange the formula to isolate \(k\). Start with the given formula \(V = k \times x^3\) and substitute known values. For example, with \( V = 36\pi \) and \( x = 3 \), the intermediate step is \( 36\pi = k \times 27 \). By dividing both sides by 27, you isolate \( k \) resulting in \( k = \frac{36\pi}{27} \). Simplify further to \( \frac{4\pi}{3} \). Such manipulations are vital in making complex equations manageable and finding precise values.