Problem 94 - Chemistry The Science in Context

Question

Ms. Good son's geology classes are popular because of their end-of-the-year field trips. Some last several days, but all involve exactly eight hours of hiking per day. On one three-day trip the class's average hiking speeds were \(1.6 \mathrm{mi} / \mathrm{h}, 1.4 \mathrm{mi} / \mathrm{h},\) and \(1.7 \mathrm{mi} / \mathrm{h}\) each day. What was the length of their trip in miles and kilometers?

Step-by-Step Solution

Answer

Answer: The length of the trip was 37.6 miles and 60.51 kilometers.

Step 1: Calculate the distance of the trip in miles for each day

To find the distance of the trip for each day, we need to multiply the average speed of each day by the total hours spent hiking each day. Day 1: \((1.6\,\text{mi}/\text{h})\times 8\,\text{h} = 12.8\,\text{mi}\) Day 2: \((1.4\,\text{mi}/\text{h})\times 8\,\text{h} = 11.2\,\text{mi}\) Day 3: \((1.7\,\text{mi}/\text{h})\times 8\,\text{h} = 13.6\,\text{mi}\) Now we have distances for each day.

Step 2: Calculate the total distance of the trip in miles

Add the distances calculated for each day to find the total distance of the trip in miles: Total distance: \(12.8\,\text{mi} + 11.2\,\text{mi} + 13.6\,\text{mi} = 37.6\,\text{mi}\)

Step 3: Convert the distance from miles to kilometers

To convert the distance from miles to kilometers, we need to know the conversion factor between miles and kilometers. This factor is given by 1 mile = 1.60934 kilometers Therefore, to convert the total distance in miles to kilometers, we have to multiply the total distance by the conversion factor: Total distance: \(37.6\,\text{mi}\times 1.60934\,\frac{\text{km}}{\text{mi}} = 60.51064\,\text{km}\)

Step 4: Round the answer

The length of their trip in kilometers to two decimal places is: Total distance: \(60.51\,\text{km}\) So, the length of the trip was 37.6 miles and 60.51 kilometers.

Key Concepts

Unit Conversion

Unit conversion is a crucial skill in mathematical problem solving, particularly when determining distances in different measurement systems. For this exercise, converting miles to kilometers involves using a specific conversion factor. To change miles into kilometers, you multiply the distance in miles by the conversion factor 1.60934.

For example, if you have a 37.6-mile journey, you calculate the kilometers by:

• 37.6 miles × 1.60934 = 60.51064 kilometers

It's essential to remember and utilize these conversion factors accurately to solve problems involving different units of measurement. This understanding allows us to easily communicate distances and understand them in varying contexts, like converting for international travel or competitions.

Average Speed

Average speed is a key concept when calculating how far a person or an object has traveled over time. It's calculated by dividing the total distance traveled by the total time taken. This concept is especially useful in settings like hiking, where the speed may vary each day.

In this problem, Ms. Good's class hiked with different average speeds each day, specifically:

• Day 1: 1.6 miles/hour
• Day 2: 1.4 miles/hour
• Day 3: 1.7 miles/hour

These speed values allow us to determine how far they travel each day (distance = speed × time), which is foundational for planning trips and estimating travel times, making it a practical application of average speed calculations.

Hiking Distance

Calculating hiking distance involves multiplying the average speed by the time spent hiking. This compiles your daily summaries into a total trip length. In Ms. Good's field trip, the class hiked eight hours each day at varying average speeds. The formula used was:

• Distance = Speed × Time

The specific distances were:

• Day 1: 1.6 mi/h × 8 h = 12.8 miles
• Day 2: 1.4 mi/h × 8 h = 11.2 miles
• Day 3: 1.7 mi/h × 8 h = 13.6 miles

Adding these up gave the total hike distance as 37.6 miles. Knowing how to calculate hiking distances helps in planning routes and ensuring safety during hiking trips.

Mathematical Problem Solving

Engaging in mathematical problem solving, like finding the hiking trip distance, enhances critical thinking and application skills. This exercise required multiple steps:

• First, calculate daily distances using average speed and time.
• Second, sum daily results to find total trip distance.
• Third, use unit conversion for a complete understanding in different measurements.

Each step develops problem-solving abilities by breaking down the task into manageable segments. This approach is not just about reaching a solution, but about understanding each layer of the problem. It provides a structured method applicable to various areas of life, encouraging methodical and logical thinking.