Problem 57 - Algebra 1

Question

You learned that bald eagles fly up to 30 miles per hour and dive at speeds up to 100 miles per hour. Using this information, write and solve an equation to answer each question. a. What is the least amount of time that an eagle could take to fly 6 miles? b. An eagle a mile above the water spots a fish. What is the shortest time it would take the eagle to dive for the fish? Express your answer in seconds.

Step-by-Step Solution

Answer

a. The eagle would take at least 12 minutes to fly 6 miles. b. The eagle would take 36 seconds to dive 1 mile for the fish.

Step 1: Determine time for flying distance

To find out how long it would take for the eagle to fly 6 miles, divide the total distance by the manner of travel's speed. In this case, the eagle is flying, so the speed is 30 miles per hour. The calculation would be: \(Time = \frac{Distance}{Speed}\) = \( \frac{6 \text{ miles}}{30 \text{ miles per hour}}\) = 0.2 hours

Step 2: Convert time into more appropriate units

The answer is in hours, while usually such short durations are expressed in minutes. To convert 0.2 hours into minutes, multiply by 60 (as there are 60 minutes in an hour). So, 0.2 hours * 60 minutes/hour = 12 minutes.

Step 3: Determine time for diving distance

To find out how long it would take for the eagle to dive a mile, divide the total distance by the manner of travel's speed. In this case, the eagle is diving, so the speed is 100 miles per hour. The calculation would be \(Time = \frac{Distance}{Speed}\) = \( \frac{1 \text{ mile}}{100 \text{ miles per hour}}\) = 0.01 hours

Step 4: Convert time into more appropriate units

The answer is in hours, but the problem is asking for the result in seconds. To convert 0.01 hours into seconds, do the following calculations: 0.01 hours * 60 minutes/hour = 0.6 minutes and 0.6 minutes * 60 seconds/minute = 36 seconds.

Key Concepts

Speed Calculations

To solve distance-time problems, calculating speed is essential. Speed determines how fast an object moves over a certain distance. In the formula for speed calculations, speed equals distance divided by time:

• Speed (\( S \)) = Distance (\( D \)) / Time (\( T \)).

In our example with the bald eagle, we focused on two scenarios: flying and diving. For the flying part, the eagle travels a distance of 6 miles at a speed of 30 miles per hour. To determine how long it takes the eagle to cover this distance, rearrange the formula to solve for time:

• \( T = \frac{D}{S} = \frac{6 \, \text{miles}}{30 \, \text{miles per hour}} = 0.2 \, \text{hours} \)

Similarly, to find out how long a dive takes, use the distance of 1 mile at a speed of 100 miles per hour:

• \( T = \frac{D}{S} = \frac{1 \, \text{mile}}{100 \, \text{miles per hour}} = 0.01 \, \text{hours} \)

Unit Conversion

Understanding unit conversion is essential, especially when calculations must be presented in different units. After calculating the time in hours, you will often need to convert it to other units, like minutes or seconds, for clearer understanding.
For our flying example, we first calculated it takes 0.2 hours to fly 6 miles. Since it's common to use minutes for short durations, convert hours into minutes by multiplying by 60:

• 0.2 hours × 60 minutes/hour = 12 minutes.

For the diving scenario, the time of 0.01 hours seemed quite small, so converting into seconds can prove more practical. Follow two steps:

• First convert hours into minutes: 0.01 hours × 60 minutes/hour = 0.6 minutes.
• Then convert these minutes into seconds: 0.6 minutes × 60 seconds/minute = 36 seconds.

These conversions help communicate the time in a more user-friendly way, allowing easy comparisons and better understanding.

Problem Solving Steps

Solving real-world problems involves clear and logical steps. Adhering to these can simplify the process and lead to accurate results.
1. **Identify the Information Given**: Carefully read the problem and note down details. In this exercise, we were provided speeds for flying and diving, and respective distances.
2. **Set Up the Equation**: Use relevant formulas based on what you're solving for. Here, we used the formula \( T = \frac{D}{S} \).
3. **Perform Calculations in Phases**: Break down the problem into smaller tasks. First, solve for the time in hours. Then, convert this time into the more suitable unit requested.
4. **Review Your Answer**: Always double-check your calculations and ensure units meet the problem's requirements. Verifying calculations ensures consistency and accuracy.
Following these organized steps can prove invaluable for solving almost any distance-time problem efficiently.