Solve each problem. See Example 2. Callie took 20 min to drive her boat upstream to water-ski at her favorite spot. Coming back later in the day, at the same boat speed, took her 15 min. If the current in that part of the river is 5 km per hr, what was her boat speed?

Verified step by step guidance

1. Define variables: Let \(b\) represent the boat's speed in still water (in km/hr). The current speed is given as 5 km/hr.
2. Express the effective speeds: When going upstream (against the current), the boat's effective speed is \(b - 5\). When going downstream (with the current), the effective speed is \(b + 5\).
3. Convert the given times from minutes to hours: 20 minutes is \(\frac{20}{60} = \frac{1}{3}\) hours, and 15 minutes is \(\frac{15}{60} = \frac{1}{4}\) hours.
4. Set up distance equations: Since the distance to the spot is the same both ways, use the formula distance = speed \(\times\) time. Let \(d\) be the distance to the spot. Then, \(d = (b - 5) \times \frac{1}{3}\) for upstream, and \(d = (b + 5) \times \frac{1}{4}\) for downstream.
5. Set the two expressions for distance equal to each other and solve for \(b\): \((b - 5) \times \frac{1}{3} = (b + 5) \times \frac{1}{4}\). Multiply both sides to clear denominators and solve the resulting linear equation for \(b\).

Key Concepts

Relative Speed in Upstream and Downstream Motion

When moving in a river with a current, the effective speed of a boat changes. Upstream speed is the boat's speed minus the current speed, while downstream speed is the boat's speed plus the current speed. Understanding this helps set up equations relating time, distance, and speed.

Distance, Speed, and Time Relationship

The fundamental formula connecting distance, speed, and time is Distance = Speed × Time. Since the distance traveled upstream and downstream is the same, this relationship allows us to create equations to solve for the unknown boat speed.

Solving Systems of Equations

To find the boat speed, you often need to solve two equations simultaneously—one for upstream and one for downstream motion. This involves algebraic manipulation to isolate and determine the unknown variable representing the boat's speed.