Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

A man rows a boat 500 feet upstream against a constant current in 10 minutes. He then rows 300 feet downstream (with the same current) in 5 minutes. Find the speed of the current and the equivalent rate at which he can row in still water.

Step-by-Step Solution

Answer

Current speed: 5 feet/minute, Rowing speed in still water: 55 feet/minute.

Step 1: Understand the Problem

We need to find two things: the speed of the current and the rowing speed in still water. We are given distances and times for rowing upstream and downstream which will be useful to form equations.

Step 2: Define Variables

Let \( c \) be the speed of the current in feet per minute, and \( r \) be the rowing speed in still water in feet per minute.

Step 3: Formulate Equations for Upstream

While rowing upstream, the effective speed is \( r - c \). He rows 500 feet in 10 minutes, so: \[ 500 = (r - c) imes 10 \]Simplify to obtain the first equation: \[ 50 = r - c \]

Step 4: Formulate Equations for Downstream

While rowing downstream, the effective speed is \( r + c \). He rows 300 feet in 5 minutes, so: \[ 300 = (r + c) imes 5 \]Simplify to obtain the second equation: \[ 60 = r + c \]

Step 5: Solve the System of Equations

We have the system of equations:\[\begin{align*}1. & \quad r - c = 50 \2. & \quad r + c = 60\end{align*}\] Add both equations to eliminate \( c \):\[(r - c) + (r + c) = 50 + 60\]\[2r = 110 \Rightarrow r = 55\]Subtract the first equation from the second:\[(r + c) - (r - c) = 60 - 50\]\[2c = 10 \Rightarrow c = 5\]

Step 6: Conclude the Solution

The speed of the current \( c \) is 5 feet per minute, and the equivalent rowing speed in still water \( r \) is 55 feet per minute.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the values of variables that make the equation true. In our exercise, we have two key equations formulated from a real-world situation. The equations are:

• Upstream: \( r - c = 50 \)
• Downstream: \( r + c = 60 \)

Finding solutions to these equations involves manipulating them to isolate the variables involved.
We often use techniques such as addition or subtraction to combine equations, making it easier to solve for unknowns. Here, we added the two equations to remove the variable \( c \), allowing us to solve for \( r \). After finding \( r \), we used it to solve for \( c \).
This systematic approach is powerful and applies to a wide range of problems beyond rowing speeds and currents.

Word Problems

Word problems can feel intimidating, but they are just real-life scenarios described with words that translate into mathematical equations. Let's break it down. Frameworks like identifying what is asked, organizing given data, and determining relationships between different elements of the problem help in converting words into equations.
The language of this exercise explains a man rowing a boat in a current, providing distances and times for two conditions: upstream and downstream.
When tackling word problems:

• Identify what you need to find; here it's the current's speed and rowing speed in still water.
• Label variables; we used \( r \) for rowing speed and \( c \) for current speed.
• Create equations from descriptions, using known formulas like distance = speed \( \times \) time.

This structured approach simplifies word problems, turning them into manageable algebraic equations.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In this problem, our goal was to find two unknowns: the current speed and the rowing speed in still water.
The system is represented as:

• Equation 1: \( r - c = 50 \)
• Equation 2: \( r + c = 60 \)

To solve, we looked for values of \( r \) and \( c \) that satisfy both equations simultaneously. One common method involves adding or subtracting equations to eliminate one variable, then solving for the remaining variable.
This process highlights the beauty of solving systems—it boils down complex problems into simpler calculations.
With practice, systems of equations become a powerful tool for modeling and solving complex real-life situations.