Problem 48 - Algebra and Trigonometry

Question

A boat on a river downstream between two points, 20 mi apart, in one hour. The return trip against the current takes 2\(\frac{1}{2}\) hours. What is the boat's speed, and how fast does the current in the river flow?

Step-by-Step Solution

Answer

Boat's speed: 14 mph, Current's speed: 6 mph.

Step 1: Understanding the Problem

We need to find two things: the speed of the boat in still water and the speed of the river current. We have two situations: one where the boat travels downstream with the current and one where it goes upstream against it.

Step 2: Defining Variables

Let the speed of the boat in still water be \( b \) mph and the speed of the current be \( c \) mph. Thus, downstream speed is \( b + c \) and upstream speed is \( b - c \).

Step 3: Writing the Equations

For the downstream trip: the boat covers 20 miles in 1 hour. Thus, \( b + c = 20 \). For the upstream trip: the boat covers 20 miles in 2.5 hours. Thus, \( \frac{20}{2.5} = b - c \).

Step 4: Solving for Variables

First, calculate the upstream speed: \( \frac{20}{2.5} = 8 \). So the second equation is \( b - c = 8 \). We now have two equations: \( b + c = 20 \) and \( b - c = 8 \).

Step 5: Solving the System of Equations

Add the two equations to eliminate \( c \): \( (b + c) + (b - c) = 20 + 8 \). This simplifies to \( 2b = 28 \), so \( b = 14 \). To find \( c \), substitute \( b = 14 \) back into the first equation: \( 14 + c = 20 \), giving \( c = 6 \).

Step 6: Conclusion

The speed of the boat in still water is 14 mph, and the speed of the river current is 6 mph.

Key Concepts

Systems of Equations

Systems of equations are a group of two or more equations with the same set of variables. These can be solved simultaneously to find the values of the variables. In our exercise, the unknown variables are the speed of the boat in still water and the speed of the river current. We set two equations based on travel scenarios: one for downstream travel where the boat moves faster due to the current, and one for upstream travel where the boat moves slower against the current.

To solve the system of equations:

• We define the equations based on given data and relationships. For downstream, the equation uses the formula for speed, which is distance over time, resulting in the equation: \( b + c = 20 \).
• Similarly, for upstream travel, dividing the traveled distance by the travel time we establish \( b - c = 8 \).
• By adding the equations, we can eliminate one variable to solve for the other. This approach is straightforward and reduces complexity.

Practicing this method is crucial for tackling various algebraic problems involving multiple variables.

Speed and Distance

Understanding the relationship between speed, distance, and time is fundamental in algebra. These three components are interconnected through the formula:

• Speed \( = \frac{\text{Distance}}{\text{Time}} \)
• Distance \( = \text{Speed} \times \text{Time} \)
• Time \( = \frac{\text{Distance}}{\text{Speed}} \)

In our boat problem, the known distances and travel times enable us to find unknown speeds. The downstream trip provides a direct calculation: 20 miles in 1 hour results in a speed of 20 mph including the current's speed.

When the boat travels upstream, it covers the same distance in 2.5 hours. Applying the speed formula again, we calculate the boat's effective speed to be 8 mph against the current. This calculation is essential to set up our equations. Understanding this basic concept allows you to solve a wide range of problems related to movement and time.

River Current Problems

River current problems often confuse students because they involve multiple speeds: the speed of the current and the speed of an object in still water. In our problem, we dealt with a boat's movement affected by river current both upstream and downstream.

Key points to remember:

• Downstream: The boat's effective speed increases by the current's speed. This is why downstream equations have the form \( b + c \).
• Upstream: The boat's effective speed decreases due to the current. This leads to upstream equations taking the form \( b - c \).
• The current itself doesn't change, just how it impacts the objects moving through it!

By analyzing how the current affects movement in each direction, you can correctly set up and solve your equations. Mastering this concept will help you with many real-world application problems involving water currents or winds.