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 the circle, and \(r\) represents the radius of the circle.
2. Recognize the constant multiplier in the formula: \$2\pi\( is a constant value that relates \)C\( and \)r$.
3. Understand the relationship type: Since \(C\) is equal to a constant times \(r\), we say that \(C\) varies directly or is proportional to \(r\).
4. Write the phrase using the word 'varies': "The circumference \(C\) varies directly as the radius \(r\)."
5. Alternatively, write the phrase using the word 'proportional': "The circumference \(C\) is proportional to the radius \(r\)."

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, meaning as the radius increases, the circumference increases proportionally.

Proportionality Constant

The proportionality constant is the fixed multiplier that relates two varying quantities in a direct variation. In the formula C = 2πr, the constant 2π links the circumference and radius, indicating that circumference is always 2π times the radius.

Translating Mathematical Formulas into English

This involves expressing mathematical relationships using words, often highlighting how variables depend on each other. Using terms like 'varies' or 'is proportional to' helps clarify the nature of the relationship, making formulas more understandable in everyday language.