Problem 26 - Algebra A Combined Function

Question

A boat can travel 9 miles upstream in the same amount of time it takes to travel 11 miles downstream. If the current of the river is 3 miles per hour, complete the chart below and use it to find the speed of the boat in still water. $$ \begin{array}{|l|cll|} \hline & \text { Distance } & = & \text { Rate } & \text { - Time } \\ \hline \text { Upstream } & 9 & & r-3 & \\ \hline \text { Downstream } & 11 & & r+3 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The speed of the boat in still water is 30 miles per hour.

Step 1: Understand the problem

The problem describes a boat traveling upstream against the current and downstream with the current. We need to find the speed of the boat in still water, given that travel times are equal for both directions.

Step 2: Define the variables and equations

Let \( r \) represent the speed of the boat in still water. The problem states that the upstream rate is \( r - 3 \) (due to the current) and the downstream rate is \( r + 3 \). The distances are given as 9 miles upstream and 11 miles downstream.

Step 3: Use the relationship between distance, rate, and time

The time it takes to travel a distance is given by the formula \( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \). Therefore, the upstream time is \( \frac{9}{r-3} \) and the downstream time is \( \frac{11}{r+3} \).

Step 4: Set up the equation

Since the times for the upstream and downstream trips are equal, we set their equations equal to each other: \( \frac{9}{r-3} = \frac{11}{r+3} \).

Step 5: Solve the equation

Cross-multiply to get: \( 9(r+3) = 11(r-3) \). This gives \( 9r + 27 = 11r - 33 \). Simplify and solve for \( r \) to find: \( 2r = 60 \), so \( r = 30 \).

Step 6: Verify the solution

Check by substituting \( r = 30 \) back into the time equations: Upstream time is \( \frac{9}{30-3} = \frac{9}{27} = \frac{1}{3} \) hours and Downstream time is \( \frac{11}{30+3} = \frac{11}{33} = \frac{1}{3} \) hours. Since they are equal, the solution is correct.

Key Concepts

Distance-Rate-Time Relationship

The relationship between distance, rate, and time is a fundamental concept in solving rate problems. This relationship is expressed through the formula \( \text{Distance} = \text{Rate} \times \text{Time} \). To find one of the values, you need to know the other two. For example, to calculate time, you rearrange the formula to \( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \).

- **Distance** is how far an object travels.
- **Rate** is the speed at which the object is moving.
- **Time** is the duration the object takes to cover that distance.

This formula helps us solve problems by allowing the manipulation of known values to find unknowns. In our exercise, understanding this relationship allows us to calculate travel times for both upstream and downstream movements based on given distances and rates.

Upstream and Downstream Movement

In water navigation, upstream and downstream movements refer to traveling against or with the current. This affects the speed of the boat.

- **Upstream movement** is against the current. Say, the boat's still water speed is \( r \), then upstream speed is \( r-3 \), because the current slows it down by 3 mph.
- **Downstream movement** is with the current. Here, speed is \( r+3 \), as the current adds to the boat's speed.

Understanding the effect of current on speed is vital in calculating accurate travel times. The boat's time in each direction can be found by dividing distance by the adjusted rate, allowing for solving complex problems like the given exercise.

Algebraic Equations

Algebraic equations are often used to solve for unknowns in math problems. For this problem, we need to find the boat's speed in still water, which is our unknown, represented by \( r \).

We know:
- Upstream time equation: \( \frac{9}{r-3} \)
- Downstream time equation: \( \frac{11}{r+3} \)

Since the travel times are equal for both upstream and downstream, we can form the equation \( \frac{9}{r-3} = \frac{11}{r+3} \). Solving this equation will help us find \( r \).

This involves cross-multiplying to eliminate the fractions, resulting in a simpler linear equation. Isolating \( r \) gives us the solution for the boat's speed in still water. Understanding these steps is crucial for solving similar algebraic rate problems.

Problem-Solving Steps

Problem-solving involves a systematic approach, which makes tackling complex rate problems easier.

**Steps for Solving Rate Problems:**
- **Understand** the problem: Identify what you need to find and what information is given.
- **Define** the variables: Assign symbols to unknown values, like \( r \) for boat speed in still water.
- **Use** formulas: Apply the distance-rate-time relationship to formulate equations.
- **Set up** equations: Based on given conditions, such as equal travel times in our problem.
- **Solve** equations: Use algebraic methods such as cross-multiplying to find the unknown.
- **Verify** the solution by plugging it back into the original equations.

These steps help solve not only river and boat problems, but any scenario involving rates, distances, and times. Practicing them enhances problem-solving skills for a wide range of mathematical challenges.