Write each formula as an English phrase using the word varies or proportional. r = d/t, where r is the speed when traveling d miles in t hours.

Verified step by step guidance

1. Identify the variables in the formula: \(r\) represents speed, \(d\) represents distance, and \(t\) represents time.
2. Recognize that the formula \(r = \frac{d}{t}\) expresses speed as the ratio of distance to time.
3. Express the relationship between speed and distance: speed \(r\) varies directly with distance \(d\) when time \(t\) is held constant.
4. Express the relationship between speed and time: speed \(r\) varies inversely with time \(t\) when distance \(d\) is held constant.
5. Combine these ideas into a phrase: Speed varies directly with distance and inversely with time.

Key Concepts

Direct Variation and Proportionality

Direct variation describes a relationship where one quantity changes in proportion to another. If y varies directly as x, then y = kx for some constant k. Understanding this helps express formulas in words using 'varies' or 'proportional to.'

Interpreting Formulas in Context

Translating mathematical formulas into English requires identifying what each variable represents and how they relate. For r = d/t, recognizing r as speed, d as distance, and t as time allows us to describe the relationship clearly in words.

Units and Rates

Speed is a rate defined as distance traveled per unit of time, typically miles per hour. Understanding rates and their units is essential to correctly interpret and phrase formulas involving ratios like r = d/t.