A rabbit travels at a constant speed. The table below shows the proportional relationship between the distance and time a rabbit travels.
What is the constant of proportionality?
Time in hours, x	Distance in miles, y
2
• ?
3
4
• ?
6
6
• ?
9
8
• ?
12
10
• ?
15


The constant of proportionality is 
.
 


To find the constant of proportionality, we need to determine the ratio of the distance traveled (in miles) to the time (in hours) for each pair of data.

We have the following pairs of time and distance from the table:
- Time = 2 hours → Distance = ?
- Time = 3 hours → Distance = 6 miles
- Time = 4 hours → Distance = ?
- Time = 6 hours → Distance = 9 miles
- Time = 8 hours → Distance = 12 miles
- Time = 10 hours → Distance = 15 miles

The constant of proportionality \( k \) is the ratio between distance and time, represented as:

\[
k = \frac{\text{Distance (y)}}{\text{Time (x)}}
\]

Using the given dataTo find the constant of proportionality, we need to determine the ratio of the distance traveled (in miles) to the time (in hours) for each pair of data.

We have the following pairs of time and distance from the table:
- Time = 2 hours → Distance = ?
- Time = 3 hours → Distance = 6 miles
- Time = 4 hours → Distance = ?
- Time = 6 hours → Distance = 9 miles
- Time = 8 hours → Distance = 12 miles
- Time = 10 hours → Distance = 15 miles

The constant of proportionality \( k \) is the ratio between distance and time, represented as:

\[
k = \frac{\text{Distance (y)}}{\text{Time (x)}}
\]

Using the given 
- For time = 3 hours and distance = 6 miles:
\[
k = \frac{6}{3} = 2
\]

- For time = 6 hours and distance = 9 miles:
\[
k = \frac{9}{6} = 1.5
\]

- For time = 8 hours and distance = 12 miles:
\[
k = \frac{12}{8} = 1.5
\]

- For time = 10 hours and distance = 15 miles:
\[
k = \frac{15}{10} = 1.5
\]

Since the ratio (constant of proportionality) is consistently \( 1.5 \) in all cases, the **constant of proportionality is 1.5**.

Would you like more details or have any questions?

Here are 5 questions related to this problem:
1. How does the constant of proportionality relate to the speed of the rabbit?
2. How would the relationship change if the rabbit slowed down over time?
3. Can you create an equation using the constant of proportionality for the rabbit’s journey?
4. If the rabbit traveled 20 miles, how long would it take given this constant speed?
5. What is the significance of proportionality in real-world applications?

**Tip:** The constant of proportionality in a relationship between two quantities tells you how much one quantity changes in relation to the other.

A rabbit travels at a constant speed. The table below shows the proportional relationship between the distance and time a rabbit travels. What is the constant of proportionality? Time in hours, x Distance in miles, y 2 • ? 3 4 • ? 6 6 • ? 9 8 • ? 12 10 • ? 15

Determine the constant of proportionality for a rabbit's journey using a proportional relationship between time and distance. Given pairs like 3 hours and 6 miles, find the constant ratio of 1.5. This lesson covers key concepts in ratios and proportional relationships for Grades 6-8.

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