Problem 48 - Algebra 1

Question

A steamboat went 8 miles upstream in 1 hour. The return trip took only 30 minutes. Assume that the speed of the current and the direction were constant during both parts of the trip. Find the speed of the boat in still water and the speed of the current.

Step-by-Step Solution

Answer

The speed of the boat in still water is 12 mph and the speed of the current is 4 mph.

Step 1: Define the Variables

Let's denote \(b\) as the speed of the boat in still water (in mph) and \(c\) as the speed of the current (in mph).

Step 2: Set Up the Equations

We know that the boat went upstream (against the current) 8 miles in 1 hour, so the effective speed of the boat is \(8 mph\). This gives us the equation \(b - c = 8\). We also know that the return trip downstream (with the current) took only 30 minutes which is 0.5 hours. Thus, the effective speed in this case was \(\frac{8 miles}{0.5 hours} = 16 mph\). This gives us the second equation \(b + c = 16\). Now we have a system of two equations to solve.

Step 3: Solve the System of Equations

We can solve this system by addition. Adding the two equations together gives us \(2b = 24\). So, \(b = 12 mph\). Substituting \(b = 12 mph\) into the first equation gives us \(c = 12 - 8 = 4 mph\).

Step 4: Interpret the Solution

The speed of the boat in still water (b) is 12 mph and the speed of the current (c) is 4 mph. This means that without the current, the boat can travel 12 miles in one hour, and the current itself moves at a speed of 4 miles per hour.

Key Concepts

Understanding the Speed of the Boat

The speed of a boat in still water is a fundamental concept when dealing with movements across a water body such as a river or a lake. Imagine the boat floating on water with no current; it moves purely based on its motor or rowing effort. This speed is constant and does not change unless affected by external forces like currents or wind.
The speed of the boat in still water, denoted by \(b\), can be thought of as the boat's true speed when unaffected by current. In our problem, we're looking to find out how fast the boat travels if no current is pushing against it or aiding it.
In typical problems of this nature, identifying the boat's speed in still water requires examining the net speeds experienced when moving with and against the current. This helps in breaking down scenarios where you have an effective speed different from the actual speed of the boat.

Decoding the Speed of the Current

Currents can significantly affect the speed of any vessel traveling across a body of water. The speed of the current, denoted as \(c\), refers to how fast the water in the river or sea moves. This motion can either aid the boat's travel when going downstream, or hinder it when moving upstream.
In our exercise, it's crucial to understand that the current is constant. This means it affects the boat's travel in the same way throughout both upstream and downstream journeys.
When calculating travel against or with the current, we adjust the boat’s speed by adding the current when downstream (boat speed plus current makes it go faster) and subtracting it when upstream (boat speed minus current reduces the effective speed). Here, knowing the current speed helps to decipher the actual navigational speed of the boat.

How to Solve Equations for Speed

Solving a system of equations enables us to find unknown values like the speed of the boat and the speed of the current. Here’s a simplified look at the process:
1. **Identify the equations:** For our example, we derived two equations from the given details: - Upstream equation: \(b - c = 8\) - Downstream equation: \(b + c = 16\)
2. **Solve simultaneously:** One effective way to solve these is the elimination method. Add both equations: - \(b - c + b + c = 8 + 16\) - Simplifies to \(2b = 24\), which means \(b = 12\) mph.
3. **Find the other variable:** Use the value of \(b\) in one of the equations to find \(c\): - Substitute \(b = 12\) into \(b - c = 8\) - So, \(12 - c = 8\), and \(c = 4\) mph.
This logical step-by-step process captures the relationship between the two variables, providing a clear method to solve such types of problems.