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 \(r = \frac{d}{t}\), where \(r\) is the speed, \(d\) is the distance traveled, and \(t\) is the time taken.
2. Recognize that speed \(r\) depends on both distance \(d\) and time \(t\); specifically, speed increases as distance increases if time is constant, and speed decreases as time increases if distance is constant.
3. Express the relationship between speed and distance: speed \(r\) varies directly (or is directly proportional) to the distance \(d\) when time \(t\) is held constant.
4. Express the relationship between speed and time: speed \(r\) varies inversely (or is inversely proportional) to the time \(t\) when distance \(d\) is held constant.
5. Combine these ideas into a phrase: speed \(r\) varies directly with distance \(d\) and inversely with time \(t\).

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 formula r = d/t, speed varies directly with distance when time is constant.

Inverse Variation

Inverse variation occurs when one quantity increases as another decreases, expressed as y = k/x. In the formula r = d/t, speed varies inversely with time when distance is constant, meaning as time increases, speed decreases.

Translating Formulas into English Phrases

Translating mathematical formulas into English involves expressing relationships using words like 'varies directly' or 'varies inversely.' For r = d/t, this means stating how speed depends on distance and time using proportional language.