Problem 296 - Elementary Algebra

Question

In the following exercises, solve. Two boats leave the same dock in Cairo at the same time. One heads north on the Mississippi River while the other heads south. The northbound boat travels four miles per hour. The southbound boat goes eight miles per hour. How long will it take them to be 54 miles apart?

Step-by-Step Solution

Answer

4.5 hours

Step 1: Determine the relative speed

When two objects move in opposite directions, their speeds add up. The northbound boat's speed is 4 miles per hour and the southbound boat's speed is 8 miles per hour. Therefore, the relative speed is the sum of both speeds: \(4 \text{ mph} + 8 \text{ mph} = 12 \text{ mph}\)

Step 2: Set up the equation

We need to determine the time it takes for the boats to be 54 miles apart. The formula to find time when distance and speed are known is: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)Based on our problem: \( \text{Time} = \frac{54 \text{ miles}}{12 \text{ mph}} \)

Step 3: Solve for time

Calculate the time by dividing the given distance by the relative speed:\( \text{Time} = \frac{54 \text{ miles}}{12 \text{ mph}} = 4.5 \text{ hours} \)

Key Concepts

Relative Speed

Relative speed is a crucial concept when solving problems involving two or more moving objects. These can include cars, boats, or trains. It refers to the combined speed of the objects in question. For example, when two objects move in opposite directions, their speeds add together. This is because each object is moving away from the other. In the exercise, one boat travels north at 4 miles per hour (mph) while the other goes south at 8 mph. Since they are moving away from each other, their relative speed is the sum of both speeds: \[4 \text{ mph} + 8 \text{ mph} = 12 \text{ mph}\].

Distance and Time

Understanding the relationship between distance and time is essential for solving many word problems. The basic formula links these two with speed: \[ \text{Distance} = \text{Speed} \times \text{Time}\]. In our exercise, we aim to determine the time taken for the two boats to be 54 miles apart. Reworking the formula to find time, we get: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]. This shows us that time is the distance divided by speed. By knowing the distance (54 miles) and the relative speed (12 mph), we can find out how long it will take them to be that far apart.

Speed Calculation

Speed calculation is the process of determining the speed of a moving object using basic arithmetic operations. This involves understanding and applying the formulas related to speed, distance, and time. In the given exercise, to find how long it takes for two boats to be 54 miles apart, we first determined the relative speed: \[4 \text{ mph} + 8 \text{ mph} = 12 \text{ mph}\] Next, we used the formula for time: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] By substituting the values, we obtained: \[ \text{Time} = \frac{54 \text{ miles}}{12 \text{ mph}} \] = 4.5 hours. This shows us that it takes 4.5 hours for the boats to be 54 miles apart.

Word Problems

Word problems can be tricky because they require translating words into mathematical equations. Here are some tips to help:

• Read the problem carefully.
• Identify and understand all given values.
• Determine what you need to find.
• Set up equations based on known formulas.

In our exercise, the problem presented involves the speeds of two boats and the distance they will be apart in a certain time. Key information includes the speeds of the boats (4 mph and 8 mph) and their movement in opposite directions. By translating these details into equations, we found the solution effectively. Remember to break the problem down step by step, making sure to double-check each calculation.