Question

sam and bobby want to know who cycled faster. the table shows the total miles sam traveled over ti
click here to view the graph.
click here to view the table.

find the unit rate (constant of proportionality) for sam.
\\(\frac{\text{distance}}{\text{time}} = \frac{14}{2} = 7\\ \frac{\text{miles}}{\text{hour}}\\)
(simplify your answer.)
find the unit rate (constant of proportionality) for bobby.
use \\((2,\square)\\) and \\((4,\square)\\) to find the constant of proportionality.

Explanation:

To find Bobby's unit rate, we need the distance values for time \( t = 2 \) and \( t = 4 \) from his graph (since the table for Sam is given, but Bobby's data is likely on the graph). Assuming the graph for Bobby shows, for example, if at \( t = 2 \) hours, distance is \( d_1 \) miles, and at \( t = 4 \) hours, distance is \( d_2 \) miles, the unit rate (constant of proportionality) \( k \) is calculated as \( k=\frac{\Delta d}{\Delta t}=\frac{d_2 - d_1}{4 - 2}=\frac{d_2 - d_1}{2} \).

But since the graph isn't visible here, let's assume a common scenario (e.g., if the graph is a line passing through (2, 12) and (4, 24)):

Step 1: Identify two points from Bobby's graph

Suppose the points are \( (2, 12) \) (time = 2 hours, distance = 12 miles) and \( (4, 24) \) (time = 4 hours, distance = 24 miles).

Step 2: Calculate the unit rate

The unit rate (constant of proportionality) is the slope of the line, given by \( k=\frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1,y_1)=(2,12) \) and \( (x_2,y_2)=(4,24) \).

\[
k=\frac{24 - 12}{4 - 2}=\frac{12}{2}=6
\]

(Note: The actual values depend on the graph. If you provide the graph's data points, we can calculate it accurately.)

For Sam, we already have:

Step 1: Use the given data for Sam

Distance = 14 miles, Time = 2 hours.

Step 2: Calculate unit rate

Unit rate \(=\frac{\text{Distance}}{\text{Time}}=\frac{14}{2}=7\) miles per hour.

Final Answers

• Sam's unit rate: \( 7 \) miles per hour.
• Bobby's unit rate (assuming the above graph points): \( 6 \) miles per hour. (Replace with actual graph data for accuracy.)

Answer:

To find Bobby's unit rate, we need the distance values for time \( t = 2 \) and \( t = 4 \) from his graph (since the table for Sam is given, but Bobby's data is likely on the graph). Assuming the graph for Bobby shows, for example, if at \( t = 2 \) hours, distance is \( d_1 \) miles, and at \( t = 4 \) hours, distance is \( d_2 \) miles, the unit rate (constant of proportionality) \( k \) is calculated as \( k=\frac{\Delta d}{\Delta t}=\frac{d_2 - d_1}{4 - 2}=\frac{d_2 - d_1}{2} \).

But since the graph isn't visible here, let's assume a common scenario (e.g., if the graph is a line passing through (2, 12) and (4, 24)):

Step 1: Identify two points from Bobby's graph

Suppose the points are \( (2, 12) \) (time = 2 hours, distance = 12 miles) and \( (4, 24) \) (time = 4 hours, distance = 24 miles).

Step 2: Calculate the unit rate

The unit rate (constant of proportionality) is the slope of the line, given by \( k=\frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1,y_1)=(2,12) \) and \( (x_2,y_2)=(4,24) \).

\[
k=\frac{24 - 12}{4 - 2}=\frac{12}{2}=6
\]

(Note: The actual values depend on the graph. If you provide the graph's data points, we can calculate it accurately.)

For Sam, we already have:

Step 1: Use the given data for Sam

Distance = 14 miles, Time = 2 hours.

Step 2: Calculate unit rate

Unit rate \(=\frac{\text{Distance}}{\text{Time}}=\frac{14}{2}=7\) miles per hour.

Final Answers

• Sam's unit rate: \( 7 \) miles per hour.
• Bobby's unit rate (assuming the above graph points): \( 6 \) miles per hour. (Replace with actual graph data for accuracy.)