Question

Write an equation that relates the quantities. The volume \(V\) of a sphere varies directly with the cube of its radius \(r .\) The constant of proportionality is \(\frac{4 \pi}{3}\)

Short answer

The volume \( V \) of a sphere is given by \[ V = \frac{4 \pi}{3} \cdot r^3 \]

Step-by-step solution

Understanding Direct Variation

When a quantity varies directly with the cube of another quantity, it means the first quantity is equal to a constant multiplied by the cube of the second quantity.

Identify Variables and Constant

Given the volume of the sphere is denoted by \( V \) and the radius by \( r \). The constant of proportionality is provided as \( \frac{4 \pi}{3} \).

Set Up the Equation

The general form for direct variation is \( V = k \cdot r^3 \), where \( k \) is the constant. Here, \( k = \frac{4 \pi}{3} \).

Write the Final Equation

Replace \( k \) with \( \frac{4 \pi}{3} \) in the general form: \[ V = \frac{4 \pi}{3} \cdot r^3 \]

Key concepts

Direct Variation

In mathematics, when we say that one quantity varies directly with another, we mean that if one quantity increases, the other does too, in a consistent way. For example, if you have a larger radius for a sphere, the volume will also be larger. Direct variation can be written mathematically as \(y = k \, x \), where \(y\) and \(x\) are the quantities, and \(k\) is the constant of proportionality. Here, the volume of a sphere varies directly with the cube of its radius. This means that the volume \(V\) is equal to a constant \(k\) times the cube of the radius \(r^3\). Understanding this relationship helps us go from knowing the radius to calculating the volume easily.

Radius

The radius \(r\) of a sphere is the distance from the center of the sphere to any point on its surface. This is a basic concept in geometry and is crucial for calculating various properties of a sphere, including its volume. In mathematical formulas, the radius is often represented by the symbol \(r\). Knowing the radius allows us to determine many other characteristics of a sphere due to its fundamental role. For example, when calculating the volume of a sphere, it's the radius that we need to cube (multiply by itself twice) before multiplying by the constant of proportionality. So always make sure you have the correct measurement for the radius before doing any further calculations.

Constant of Proportionality

A constant of proportionality is a constant value that relates two proportional quantities. In this exercise, the volume \(V\) of a sphere is directly proportional to the cube of its radius \(r^3\). The specific constant of proportionality here is \(\frac{4\pi}{3}\), which means every time we know the radius, we can cube it and multiply by this constant to get the volume. This constant simplifies the calculation and is derived from geometric principles, providing a way to move smoothly between radius and volume. Whenever we see a direct variation like this, the constant of proportionality is what ties the two quantities together in a meaningful way.

Volume Formula

The formula for the volume of a sphere is a specific instance of a direct variation equation. In this case, the volume \(V\) is given by \(V = \frac{4\pi}{3} \cdot r^3\). This formula comes from integrating the concept of how space is occupied within a three-dimensional object with a spherical shape. To use this formula, you only need to know the radius of the sphere. Plug the radius into the formula where the radius symbol \(r\) is, cube the radius, and then multiply by the constant \(\frac{4\pi}{3}\). This elegant formula encapsulates the direct variation relationship between a sphere's volume and the cube of its radius. It's a handy tool for quickly finding the volume when the radius is known.