Set up the appropriate quadratic equations and solve. A...

Set up the appropriate quadratic equations and solve.
A hydrofoil made the round-trip of $120 \mathrm{km}$ between two islands in 3.5 h of travel time. If the average speed going was $10 \mathrm{km} / \mathrm{h}$ less than the average speed returning, find these speeds.

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Key Concepts

Distance-Speed-Time Relationship

This concept deals with the fundamental relationship between distance, speed, and time, usually expressed in the formula distance = speed × time. In problems involving travel, calculating one of these quantities involves rearranging the formula, making it essential for modeling scenarios such as round-trip journeys where different segments are covered at different speeds.

Quadratic Equations

Quadratic equations arise when the relationship between variables involves squared terms. In word problems, they commonly appear after combining relationships such as distance, speed, and time, especially when differences or sums of speeds or times lead to equations that are not linear. Solving these equations allows us to find the unknown values representing quantities like speed or time.

Formulating and Solving Word Problems

This key concept involves translating a real-world scenario into a mathematical model by setting up equations based on the given information. The process includes identifying known quantities, defining variables for unknowns, and using relationships like the distance-speed-time formula to derive equations. In many cases, such as round-trip scenarios, this leads to a quadratic equation requiring a systematic approach to solve.

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