Problem 17 - Algebra Form and Function

Question

In Exercises \(17-20,\) which letters stand for variables and which for constants? $$ V(r)=(4 / 3) \pi r^{2} $$

Step-by-Step Solution

Answer

Answer: In the given formula, 'r' represents the variable as it stands for the radius of the sphere. The constants are '4/3' and '\(\pi\)'.

Step 1: Identify the variable

In the given formula, the variable is represented by the letter 'r'. 'r' stands for the radius of the sphere, and since the volume depends on the radius, it can take on various values. Therefore, 'r' is the variable in this formula.

Step 2: Identify the constants

In the formula, we have the following constants: 1. The number '4/3', which is a fractional constant and part of the formula for finding the volume of a sphere. 2. The symbol '\(\pi\)', which is a mathematical constant, approximately equal to 3.14, that represents the ratio of a circle's circumference to its diameter. Thus, in the formula $$V(r) = (4/3)\pi r^2$$, 'r' is the variable, and '4/3' and '\(\pi\)' are constants.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operators (like addition, subtraction, multiplication, and division). It's like a sentence in mathematics that conveys an idea or relationship.
In the expression \(V(r) = (4/3)\pi r^2\), each part plays a specific role:

• **Variables:** These are symbols, often letters, that represent unknown or changeable values. In our example, 'r' is a variable that denotes the radius of the sphere. Variables can vary and take on different values, which means they are not fixed.
• **Constants:** These are fixed, unchanging values. In the expression, '4/3' and '\(\pi\)' are constants. They hold a specific value and don't change regardless of the variable's value.
• **Operators and Functions:** These include the equal sign, multiplication, and division which define how the variables and constants interact with each other. Here, the expression calculates the volume by multiplying and raising to power.

Algebraic expressions are foundational in mathematics because they allow for the formulation of equations and the solving of numerous mathematical problems.

Mathematical Constants

Mathematical constants are special numbers that arise naturally in mathematics and don't change. They are values known to have fixed numeric values.
In our expression, two constants are present:

• **The fraction \(4/3\):** This is part of the volume formula for a sphere. It's a rational constant, meaning it can be expressed as the ratio of two integers.
• **Pi (\(\pi\)):** This is one of the most famous mathematical constants. Approximately equal to 3.14159, it represents the ratio of the circumference of a circle to its diameter. Its nature is irrational, meaning it cannot be expressed as an exact fraction and its decimal form goes on forever without repeating.

These constants play a critical role in formulae and mathematical calculations, providing fixed reference points in an otherwise variable world. Understanding them helps in comprehending broader mathematical concepts and in solving complex problems.

Volume of a Sphere

The volume of a sphere refers to the amount of three-dimensional space it occupies. Calculating this volume involves an established formula, which is crucial knowledge in geometry and practical applications.

• **Formula:** The formula for the volume of a sphere is given by \(V = \frac{4}{3}\pi r^3\). This formula shows how the radius influences the volume.
• **Understanding the Parts:** The constant fraction \(4/3\) and \(\pi\) are multiplied by the cube of the radius \(r\), indicating the significance of the radius in determining the size of the sphere's volume.
• **Application:** By knowing just the radius, one can calculate how much space a sphere will occupy. This is vital in fields such as engineering, physics, and astronomy where space and volume considerations are essential.

The volume formula highlights the beautiful interplay of constants and variables in mathematics, translating abstract equations into tangible real-world measurements.