Write each formula as an English phrase using the word varies or proportional. C=2πr, where C is the circumference of a circle of radius r.

Verified step by step guidance

1. Identify the variables in the formula: \(C\) represents the circumference of a circle, and \(r\) represents the radius of the circle.
2. Recognize the constant in the formula: \(2\pi\) is a constant multiplier, where \(\pi\) is approximately 3.14159.
3. Understand the relationship: Since \(C = 2\pi r\), the circumference \(C\) changes as the radius \(r\) changes, multiplied by the constant \(2\pi\).
4. Express the relationship using the word 'varies': We say that the circumference \(C\) varies directly as the radius \(r\).
5. Write the full English phrase: 'The circumference of a circle varies directly (or is proportional) to its radius, with the constant of proportionality being \(2\pi\).'

Key Concepts

Direct Variation

Direct variation describes a relationship where one quantity changes proportionally with another. If y varies directly as x, then y = kx for some constant k. In the given formula, circumference varies directly with the radius.

Proportionality Constant

The proportionality constant is the fixed multiplier that relates two varying quantities. In the formula C = 2πr, the constant 2π shows how many times the radius is multiplied to get the circumference, indicating the strength of the proportional relationship.

Translating Mathematical Formulas into Words

This involves expressing algebraic equations using clear English phrases. For example, 'C = 2πr' can be stated as 'The circumference varies directly as the radius, with 2π as the constant of proportionality.' This skill helps in understanding and communicating mathematical relationships.