Problem 44 - College Algebra

Question

What is the total distance that two people travel in 3 \(\mathrm{h}\) if one of them is riding a bike at 15 \(\mathrm{mi} / \mathrm{h}\) and the other is walking at 3 \(\mathrm{mi} / \mathrm{h}\) ?

Step-by-Step Solution

Answer

The total distance traveled by both is 54 miles.

Step 1: Calculate distance traveled by the biker

To find the distance traveled by the person on the bike, use the formula: \( \text{distance} = \text{speed} \times \text{time} \). Here, the speed is 15 mi/h, and the time is 3 hours. So, the distance is \( 15 \times 3 = 45 \text{ miles} \).

Step 2: Calculate distance traveled by the walker

Similarly, calculate the distance for the person walking. Use the formula: \( \text{distance} = \text{speed} \times \text{time} \). With a speed of 3 mi/h for 3 hours, the distance is \( 3 \times 3 = 9 \text{ miles} \).

Step 3: Add distances to find the total distance traveled

Add the distance traveled by the biker (45 miles) and the walker (9 miles) to find the total distance: \( 45 + 9 = 54 \text{ miles} \).

Key Concepts

Distance Calculation

When it comes to the concept of distance calculation, it's all about figuring out how far someone or something has traveled over a period of time. This is usually done using simple multiplication. Take the speed at which someone is moving, which is given in units like miles per hour (mi/h), and multiply it by the time spent moving, typically in hours. The result will give you the distance covered in that specific timeframe.

In the problem mentioned above, one person rides a bike at 15 mi/h for a period of 3 hours. To find out how far the biker has traveled, calculate using the formula:

• Distance = Speed × Time
• 15 mi/h × 3 h = 45 miles

Similarly, if another person is walking at 3 mi/h for the same time of 3 hours, their distance can be calculated using the same steps:

• Distance = Speed × Time
• 3 mi/h × 3 h = 9 miles

Distance calculation is a fundamental concept that helps us determine how far an object has traveled based on how fast it moves over a specific period.

Speed and Time Relationship

The relationship between speed and time is crucial in determining distance. Speed tells us how fast an object is moving. It's usually given in units like miles per hour, kilometers per hour, or meters per second. Time, on the other hand, indicates how long the object has been in motion, often measured in hours, minutes, or seconds.

Think of speed as a measure of how much distance is covered in a single unit of time. By multiplying speed by time, we can find out how far something has traveled in total. The faster an object moves, the more distance it will cover over any given period of time. Conversely, the more time an object spends moving, at any constant speed, the further it will travel.

In our exercise, both individuals travel for the same duration of 3 hours, but at different speeds. The biker, moving faster at 15 mi/h, covers more distance compared to the walker at 3 mi/h. Being able to understand and apply this relationship allows us to predict and calculate how travel duration and speed affects distance covered.

Mathematical Formulas

Mathematical formulas are tools that help simplify and solve everyday problems like calculating travel distances. One such formula used in the exercise is:

• Distance = Speed × Time

Here, it's essential to substitute the correct values of speed and time to ensure accurate results. The formula acts like a bridge connecting all three elements of travel problems – making it easier to solve and verify results.

Let's break it down: First identify the speed, which is how fast someone or something is moving. Then, identify the time, which is how long the movement continues. Multiply these two numbers, and you have your distance. The simplicity and reliability of formulas allow us to tackle more complex problems step by step, ensuring that the logic applied is consistent in various situations.

Mastering the use of mathematical formulas builds a strong foundation for solving more advanced problems in algebra and beyond, making them indispensable in various fields from physics to everyday travel calculations.