A graph titled Fast Freddy has time (minutes) on the x-axis and Distance (miles) on the y-axis. A line goes through points (0, 0) and (4, 1). A 2-column table with 3 rows titled Speed Sam. Column 1 is labeled Time (minutes) x with entries 4, 20, 32. Column 2 is labeled Distance (miles) y with entries one-half, 2 and one-half, 4.

Find the slope from the graph and from the table to compare the speeds of two marathon runners.

What is the constant speed of Fast Freddy?

What is the constant speed of Speedy Sam?

Who ran faster?

1 answer

To determine the constant speeds of Fast Freddy and Speedy Sam, we'll need to find the slope of the line for Fast Freddy from the graph and the slope based on the data in the table for Speedy Sam.

Constant Speed of Fast Freddy

The slope \( m \) of a line can be calculated using the formula: \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points given for Fast Freddy, (0, 0) and (4, 1):

  • \( x_1 = 0 \), \( y_1 = 0 \)
  • \( x_2 = 4 \), \( y_2 = 1 \)

Now, substituting the values into the slope formula: \[ m = \frac{1 - 0}{4 - 0} = \frac{1}{4} \]

Thus, the constant speed of Fast Freddy is: \[ 0.25 \text{ miles per minute} \]

Constant Speed of Speedy Sam

From the table provided for Speedy Sam, we can calculate the slope from two pairs of points. Let's use the first two points and then the last two points:

  1. Between (4, 0.5) and (20, 2.5):

    • \( x_1 = 4 \), \( y_1 = 0.5 \)
    • \( x_2 = 20 \), \( y_2 = 2.5 \)

    Substituting into the slope formula: \[ m = \frac{2.5 - 0.5}{20 - 4} = \frac{2}{16} = \frac{1}{8} \]

    The speed from these points would be: \[ 0.125 \text{ miles per minute} \]

  2. Between (20, 2.5) and (32, 4):

    • \( x_1 = 20 \), \( y_1 = 2.5 \)
    • \( x_2 = 32 \), \( y_2 = 4 \)

    Substituting into the slope formula: \[ m = \frac{4 - 2.5}{32 - 20} = \frac{1.5}{12} = \frac{1}{8} \]

    The speed from these points would also be: \[ 0.125 \text{ miles per minute} \]

Comparing Speeds

  • Constant speed of Fast Freddy: 0.25 miles per minute
  • Constant speed of Speedy Sam: 0.125 miles per minute

Conclusion

Fast Freddy ran faster than Speedy Sam because 0.25 miles per minute (Fast Freddy's speed) is greater than 0.125 miles per minute (Speedy Sam's speed).