A motorboat heads upstream on a river that has a current of...

Name: Problem 26, Chapter 1: College Algebra
Uploaded: 2021-06-19T15:18:16-08:00
Duration: 3 min
Channel: Kyle Christian
Description: A motorboat heads upstream on a river that has a current of 3 miles per hour. The trip upstream takes 5 hours, and the return trip takes 2.5 hours. What is the speed of the motorboat? (Assume that the motorboat maintains a constant speed relative to the water.)

A motorboat heads upstream on a river that has a current of 3 miles per hour. The trip upstream takes 5 hours, and the return trip takes 2.5 hours. What is the speed of the motorboat? (Assume that the motorboat maintains a constant speed relative to the water.)

Key Concepts

Systems of Equations

By setting up equations based on the distance-time relationship for both the upstream and downstream trips, we form a system of equations with the boat’s speed relative to the water as an unknown. Solving this system is necessary to determine the unknown variable, making this a vital algebraic method in solving such problems.

Distance-Time Relationships

The distance-time relationship, given by the formula distance = speed × time, is fundamental in these types of problems. Since the distance traveled upstream and downstream is the same, this relationship allows us to establish equations that equate the distance covered with the respective effective speeds and travel times.

Relative Motion in a Moving Medium

This concept involves understanding that the observed speed of an object moving in a fluid (like water) is a combination of its own speed relative to the fluid and the speed of the fluid itself. In problems involving currents, the effective speed of the object is altered depending on whether it is moving with or against the flow.

Effective Speed Upstream and Downstream

A key principle here is that when moving upstream, the effective speed of the boat is its speed relative to the water minus the speed of the current, while downstream it is the sum of the boat’s speed and the current. This difference in effective speed directly affects the time taken for the journey in each direction and is essential for setting up the mathematical model.

Transcript

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