Problem 8 - Intermediate Algebra

Question

A boat can cruise at 30 mph in still water. a. What is its cruising speed upstream against a current of \(4 \mathrm{mph} ?\) b. What is its cruising speed downstream with a current of \(4 \mathrm{mph} ?\)

Step-by-Step Solution

Answer

a. 26 mph upstream; b. 34 mph downstream.

Step 1: Understanding The Problem

We need to determine the boat's effective cruising speeds both upstream and downstream when there is a current of 4 mph. The boat's speed in still water is given as 30 mph.

Step 2: Calculate Upstream Speed

When moving upstream, the boat's speed is reduced by the speed of the current because it works against the boat's direction. Therefore the upstream speed is calculated as follows: \( ext{Upstream Speed} = ext{Still Water Speed} - ext{Current Speed} = 30 ext{ mph} - 4 ext{ mph} = 26 ext{ mph} \).

Step 3: Calculate Downstream Speed

When moving downstream, the boat benefits from the current, effectively increasing its speed. Thus, the downstream speed is given by: \( ext{Downstream Speed} = ext{Still Water Speed} + ext{Current Speed} = 30 ext{ mph} + 4 ext{ mph} = 34 ext{ mph} \).

Key Concepts

Speed Calculation

Calculating speed is a fundamental concept in solving word problems related to motion. It is defined as the distance covered per unit of time. The formula for speed is:

• Speed = Distance ÷ Time

In the context of the boat, we will focus on the boat's cruise speed in both still and moving water. The boat's speed in still water is provided as 30 mph, representing its velocity unaffected by any current. For each scenario in the problem, this baseline speed will be adjusted based on the current's effect.

Upstream and Downstream Problems

When solving upstream and downstream problems, it's crucial to understand how a current affects movement. These problems usually involve a boat or an object moving through water.

• **Upstream:** Moving against the current, which reduces speed.
• **Downstream:** Moving with the current, which increases speed.

In this exercise, to calculate upstream speed, you reduce the current's speed from the boat's speed in still water:\[\text{Upstream Speed} = \text{Still Water Speed} - \text{Current Speed}\]
For downstream calculations, add the speed of the current to the boat's still-water speed:\[\text{Downstream Speed} = \text{Still Water Speed} + \text{Current Speed}\] Understanding these dynamics helps solve various motion-based problems, especially in algebraic word problems.

Current Speed Effect

The term 'current speed' refers to a continuous movement of water in one direction, which can significantly impact the speed of a vessel. A current can either help the boat move faster or slow it down, depending on its direction relative to the boat's.

• Increases boat speed when moving downstream.
• Decreases boat speed when moving upstream.

For example, if the current is 4 mph, it affects a boat moving upstream by reducing its effective speed by 4 mph. Conversely, while going downstream, it adds 4 mph to the boat's speed. Thus, understanding and calculating the effect of the current is essential for accurate speed calculations and can dramatically influence travel time and fuel calculations.