Question
Going Round-Trip with the World’s Fastest Land Animal
According to the Travel Almanac, the world’s fastest land animal is the cheetah. It can travel at up to 70 mph. Think of this scenario: A cheetah sitting under a tree sprints toward its prey at 70 mph. It runs back to its initial spot by the tree at a modest 40 mph. The cheetah has embarked on a round-trip. Going from point A to point B, the cheetah traveled at an average rate of 70 mph. Returning to point A, the cheetah traveled at an average rate of 40 mph.
Can we say that this cheetah’s average rate was 55 mph?
That’s one of the things you’ll determine as you work to complete this task. Make a conjecture. What do you think the answer will be?
Directions:
Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated and to insert images of graphs when needed. Make sure that all graphs or screenshots include appropriate information, such as titles and labeled axes. If your word processing program has an equation editor, you can insert your equations here. Otherwise, print this activity sheet and write your answers by hand.
You will be graded on the work you show, or your solution process, in addition to your answers. Make sure to show all of your work and to answer each question as you complete the task. Type all of your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations.
Your teacher will give you further directions as to how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or hand in a hard copy.
Before you begin, recall the formula that relates distance, rate, and time.
The distance formula is given by d = rt, where:
d represents distance. Common units are miles or feet.
t represents time. Common units are hours or seconds.
r represents rate or velocity. Common units are miles per hour or feet per second.
You can calculate distance by finding the product of an average rate of an object traveling and the duration of time for which it travels. Remember, units for distance and time must agree with the units
for rate.
According to the Travel Almanac, the world’s fastest land animal is the cheetah. It can travel at up to 70 mph. Think of this scenario: A cheetah sitting under a tree sprints toward its prey at 70 mph. It runs back to its initial spot by the tree at a modest 40 mph. The cheetah has embarked on a round-trip. Going from point A to point B, the cheetah traveled at an average rate of 70 mph. Returning to point A, the cheetah traveled at an average rate of 40 mph.
Can we say that this cheetah’s average rate was 55 mph?
That’s one of the things you’ll determine as you work to complete this task. Make a conjecture. What do you think the answer will be?
Directions:
Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated and to insert images of graphs when needed. Make sure that all graphs or screenshots include appropriate information, such as titles and labeled axes. If your word processing program has an equation editor, you can insert your equations here. Otherwise, print this activity sheet and write your answers by hand.
You will be graded on the work you show, or your solution process, in addition to your answers. Make sure to show all of your work and to answer each question as you complete the task. Type all of your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations.
Your teacher will give you further directions as to how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or hand in a hard copy.
Before you begin, recall the formula that relates distance, rate, and time.
The distance formula is given by d = rt, where:
d represents distance. Common units are miles or feet.
t represents time. Common units are hours or seconds.
r represents rate or velocity. Common units are miles per hour or feet per second.
You can calculate distance by finding the product of an average rate of an object traveling and the duration of time for which it travels. Remember, units for distance and time must agree with the units
for rate.
Answers
GPT-4o mini
To determine the cheetah's average speed during its round trip, we can follow these steps:
### Step 1: Define the Variables
Let:
- \( d \) = one-way distance to the prey (in miles)
- \( r_1 \) = speed while going to the prey = 70 mph
- \( r_2 \) = speed while returning to the tree = 40 mph
### Step 2: Calculate the Time for Each Leg of the Trip
Using the formula \( t = \frac{d}{r} \):
1. **Going to Prey**:
\[
t_1 = \frac{d}{r_1} = \frac{d}{70}
\]
2. **Returning to the Tree**:
\[
t_2 = \frac{d}{r_2} = \frac{d}{40}
\]
### Step 3: Calculate Total Time for Round Trip
The total time for the round trip is:
\[
t_{total} = t_1 + t_2 = \frac{d}{70} + \frac{d}{40}
\]
To add these fractions, we find a common denominator, which is 280:
\[
t_{total} = \frac{4d}{280} + \frac{7d}{280} = \frac{11d}{280}
\]
### Step 4: Calculate Total Distance for Round Trip
The total distance traveled for the round trip (from tree to prey and back) is:
\[
d_{total} = d + d = 2d
\]
### Step 5: Calculate Average Speed
Average speed is calculated using the formula:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
\]
Substituting the total distance and total time:
\[
\text{Average Speed} = \frac{2d}{\frac{11d}{280}} = 2d \times \frac{280}{11d} = \frac{560}{11}
\]
Calculating the division:
\[
\frac{560}{11} \approx 50.91 \text{ mph}
\]
### Step 6: Conclusion
The cheetah’s average speed for the entire round trip is approximately \( 50.91 \) mph, which is **not** 55 mph.
### Summary
- Going to prey was at 70 mph, while returning it was at 40 mph.
- The average speed for the round trip is approximately **50.91 mph**. Hence, we cannot say that the cheetah's average rate was 55 mph.
This analysis can be presented in a structured format to meet the assignment requirements, including a proper explanation and calculations as shown above. If graphs are needed, an illustration of distance vs. time for each leg of the trip could be included to visualize the speed differences.
### Step 1: Define the Variables
Let:
- \( d \) = one-way distance to the prey (in miles)
- \( r_1 \) = speed while going to the prey = 70 mph
- \( r_2 \) = speed while returning to the tree = 40 mph
### Step 2: Calculate the Time for Each Leg of the Trip
Using the formula \( t = \frac{d}{r} \):
1. **Going to Prey**:
\[
t_1 = \frac{d}{r_1} = \frac{d}{70}
\]
2. **Returning to the Tree**:
\[
t_2 = \frac{d}{r_2} = \frac{d}{40}
\]
### Step 3: Calculate Total Time for Round Trip
The total time for the round trip is:
\[
t_{total} = t_1 + t_2 = \frac{d}{70} + \frac{d}{40}
\]
To add these fractions, we find a common denominator, which is 280:
\[
t_{total} = \frac{4d}{280} + \frac{7d}{280} = \frac{11d}{280}
\]
### Step 4: Calculate Total Distance for Round Trip
The total distance traveled for the round trip (from tree to prey and back) is:
\[
d_{total} = d + d = 2d
\]
### Step 5: Calculate Average Speed
Average speed is calculated using the formula:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
\]
Substituting the total distance and total time:
\[
\text{Average Speed} = \frac{2d}{\frac{11d}{280}} = 2d \times \frac{280}{11d} = \frac{560}{11}
\]
Calculating the division:
\[
\frac{560}{11} \approx 50.91 \text{ mph}
\]
### Step 6: Conclusion
The cheetah’s average speed for the entire round trip is approximately \( 50.91 \) mph, which is **not** 55 mph.
### Summary
- Going to prey was at 70 mph, while returning it was at 40 mph.
- The average speed for the round trip is approximately **50.91 mph**. Hence, we cannot say that the cheetah's average rate was 55 mph.
This analysis can be presented in a structured format to meet the assignment requirements, including a proper explanation and calculations as shown above. If graphs are needed, an illustration of distance vs. time for each leg of the trip could be included to visualize the speed differences.