Problem 82
Question
If you average 30 \(\mathrm{mi} / \mathrm{h}\) on a 150 -mi trip and then return over the same 150 \(\mathrm{mi}\) at the rate of \(50 \mathrm{mi} / \mathrm{h},\) what is your average speed for the trip? Give reasons for your answer. (Source: David H.
Step-by-Step Solution
Verified Answer
The average speed for the trip is 37.5 mi/h.
1Step 1: Understand the Formula for Average Speed
To find the average speed for a round trip, you can't simply average the two speeds. Instead, average speed is calculated as total distance divided by total time.
2Step 2: Calculate Total Distance
The total distance for the trip is the sum of the distances for both legs of the journey. Since the trip out and back is 150 miles each way, the total distance is 150 + 150 = 300 miles.
3Step 3: Calculate Time for Each Leg of the Trip
Time for the outward journey is distance divided by speed: \[ \text{Time}_1 = \frac{150}{30} = 5 \text{ hours} \]Time for the return journey is:\[ \text{Time}_2 = \frac{150}{50} = 3 \text{ hours} \]
4Step 4: Calculate Total Time
The total time for the journey is the sum of the times for the outward and return trips:\[ \text{Total Time} = 5 + 3 = 8 \text{ hours} \]
5Step 5: Calculate Average Speed
Average speed for the entire trip is total distance divided by total time:\[ \text{Average Speed} = \frac{300}{8} = 37.5 \text{mi/h} \]
Key Concepts
Distance and Time RelationshipSpeed FormulaProblem-Solving Strategy
Distance and Time Relationship
When discussing movement, distance and time share a fundamental relationship that helps us understand how objects travel from one point to another. Simply put, distance refers to how far something has to travel. Time is the duration it takes to cover that distance. These two factors are deeply connected in journey calculations. For instance, if you're traveling on a 150-mile trip and then heading back over the same path, you're actually covering a total distance of 300 miles. This is because you need to consider both legs of your journey.
Understanding this connection is crucial because any change in distance directly impacts the time taken, assuming speed stays constant. Conversely, any alteration in time taken – like taking a break – will affect your understanding of speed when distance remains the same.
By comprehending how distance and time interrelate, you're better equipped to solve problems related to movement and travel. This understanding forms the foundation for calculating travel-related quantities, such as speed.
Understanding this connection is crucial because any change in distance directly impacts the time taken, assuming speed stays constant. Conversely, any alteration in time taken – like taking a break – will affect your understanding of speed when distance remains the same.
By comprehending how distance and time interrelate, you're better equipped to solve problems related to movement and travel. This understanding forms the foundation for calculating travel-related quantities, such as speed.
Speed Formula
The speed formula is an essential tool in travel-related problem-solving. It helps in determining how fast something is moving over a specific distance for a given time. The simple formula to remember is: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]This formula tells us that speed is the rate at which an object covers a distance. In other words, it's the measure of how quickly you're moving from one point to another.
In the context of our problem, the outward trip has a speed of 30 mi/h for 150 miles, taking 5 hours. Meanwhile, the return trip has a speed of 50 mi/h, covering the same 150 miles but in just 3 hours. By applying the speed formula, you can understand how speed impacts travel time for a given distance.
This basic formula is immensely useful in real-world scenarios, such as planning travel time or determining fuel efficiency. It allows you to compare how different speeds affect your overall travel experience, leading to more effective planning and decision-making.
In the context of our problem, the outward trip has a speed of 30 mi/h for 150 miles, taking 5 hours. Meanwhile, the return trip has a speed of 50 mi/h, covering the same 150 miles but in just 3 hours. By applying the speed formula, you can understand how speed impacts travel time for a given distance.
This basic formula is immensely useful in real-world scenarios, such as planning travel time or determining fuel efficiency. It allows you to compare how different speeds affect your overall travel experience, leading to more effective planning and decision-making.
Problem-Solving Strategy
Handling problems related to average speed can seem complex at first, but with an effective problem-solving strategy, it becomes manageable.
- Understand the Goal: The objective is to find the average speed for the entire trip, not just average the two speeds provided.
- Break it Down: First, calculate the total distance (300 miles) and then work out the time taken for each part of the journey – 5 hours out, 3 hours back.
- Form the Equation: Use the formula for average speed which is total distance divided by total time: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- Calculate: Plug in the values from your calculations: \[ \text{Average Speed} = \frac{300}{8} = 37.5 \text{ mi/h} \]
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