Problem 29 - Beginning and Intermediate Algebra

Question

Solve each problem. Round answers to the nearest tenth as needed. On a windy day William found that he could travel 16 mi downstream and then 4 mi back upstream at top speed in a total of 48 min. What was the top speed of William's boat if the rate of the current was \(15 \mathrm{mph} ?\) (Let \(x\) represent the rate of the boat in still water.)

Step-by-Step Solution

Answer

William's boat speed in still water is approximately 22.5 mph.

Step 1: Define the given information

William travels downstream and upstream with a current rate of 15 mph. Represent the rate of the boat in still water as \( x \) mph.

Step 2: Write the expressions for the boat's effective speeds

The effective speed downstream (with the current) is \( x + 15 \) mph, and the effective speed upstream (against the current) is \( x - 15 \) mph.

Step 3: Calculate the time taken for each segment

The time taken to travel downstream 16 mi is \( \frac{16}{x+15} \) hours. The time taken to travel upstream 4 mi is \( \frac{4}{x-15} \) hours.

Step 4: Setup the equation using the total time

The total travel time is the sum of the times for each leg of the trip, which equals 48 minutes (or \( \frac{48}{60} = 0.8 \) hours). Thus, the equation is \( \frac{16}{x+15} + \frac{4}{x-15} = 0.8 \).

Step 5: Solve the equation for \( x \)

Solve \( \frac{16}{x+15} + \frac{4}{x-15} = 0.8 \) by finding a common denominator and combining the fractions. Multiply through by \( (x+15)(x-15) \) to clear the fractions.

Step 6: Simplify and solve the resulting quadratic equation

After clearing the fractions and simplifying, solve the resulting quadratic equation to find \( x \).

Step 7: Check the solution and round to the nearest tenth

Verify the solution by substituting back into the equation and round to the nearest tenth as needed.

Key Concepts

Algebraic Problem Solving

Solving algebraic problems involves breaking down the problem into smaller, more manageable steps. In this exercise, the problem is to determine the top speed of William's boat in still water given certain conditions. We start by defining variables, in this case, letting \( x \) represent the boat's speed in still water. Next, we express the boat's effective speeds both downstream and upstream by accounting for the current's influence. These steps form the foundation of setting up the mathematical equation we need to solve. Once the equation is established, we isolate the variable \( x \) and solve it algebraically, often involving step-by-step simplification and manipulation of a quadratic equation. The final step involves verifying our solution to ensure accuracy.

Word Problems

Word problems require translating a text-based scenario into a mathematical equation. The key is identifying relevant information and formulating it into expressions or equations. In the given problem, the scenario describes William's travel times and distances downstream and upstream. Important details include the distance traveled (16 miles downstream and 4 miles upstream), the total time taken (48 minutes or 0.8 hours), and the current speed (15 mph). These details help us set up expressions for the boat's effective speed and ultimately derive an equation to solve. Carefully reading the problem, identifying the variables, and setting up corresponding equations are crucial for solving word problems effectively.

Rates and Speed Calculations

Understanding rates and speed calculations is essential in problems involving motion. For William's boat, the key is knowing how to calculate the effective speed downstream and upstream. When traveling downstream, the boat's effective speed is its speed in still water plus the current's speed (\( x + 15 \) mph). Conversely, traveling upstream against the current results in a reduced effective speed (\( x - 15 \) mph). Using these speeds, we calculate the time taken for each segment of the trip by dividing the distance by the effective speed: \( \frac{16}{x+15} \) for downstream and \( \frac{4}{x-15} \) for upstream. Knowing these basic calculations aids in setting up the proper equations to solve for the unknown variables.

Downstream and Upstream Motion

Understanding downstream and upstream motion is crucial for solving problems involving currents or winds. In the presented problem, William's boat encounters a downstream current that aids his movement and an upstream current that resists it. When working with such scenarios, it's important to consider how the current affects the boat's speed. The downstream speed (\( x + 15 \)) combines the boat's speed with the current's speed, facilitating faster travel. On the other hand, the upstream journey subtracts the current's speed from the boat's speed (\( x - 15 \)), slowing it down. These concepts help us understand the dynamics of motion in a flowing medium and are essential for setting up the correct equations to solve such problems.