To derive a formula from the given data, we can observe the relationship between time (x) and distance (y).
From the table, we can summarize the data as follows:
- When \( x = 1 \), \( y = 75 \)
- When \( x = 2 \), \( y = 150 \)
- When \( x = 3 \), \( y = 225 \)
- When \( x = 4 \), \( y = 300 \)
We can see that as time \( x \) increases, the distance \( y \) also increases. The pattern demonstrates a linear relationship.
To find the formula, let’s first determine the rate of change (slope) between the points:
The distance increased by 75 feet for each additional minute (from 75 to 150 feet, etc.).
Now, this gives us a slope \( m = 75 \) feet/minute.
The general formula for a linear relationship is:
\[ y = mx + b \]
Where:
- \( m \) is the slope,
- \( b \) is the y-intercept.
To find \( b \), we can substitute one of the points into the equation. Let's use the point \( (1, 75) \):
\[ 75 = 75(1) + b \] \[ 75 = 75 + b \] \[ b = 75 - 75 \] \[ b = 0 \]
Thus, the formula is:
\[ y = 75x \]
This means that for every minute of time \( x \), the distance \( y \) will be \( 75 \times x \) feet.
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