To create a formula for the relationship between time (x) and distance (y) based on the given data, we can observe the pattern in the table.
The data points are as follows:
- When \( x = 1 \), \( y = 75 \)
- When \( x = 2 \), \( y = 150 \)
- When \( x = 3 \), \( y = 225 \)
- When \( x = 4 \), \( y = 300 \)
To find the formula, we can first identify the slope of the relationship:
- Calculate the change in y for each change in x:
- From \( x = 1 \) to \( x = 2 \): \( 150 - 75 = 75 \)
- From \( x = 2 \) to \( x = 3 \): \( 225 - 150 = 75 \)
- From \( x = 3 \) to \( x = 4 \): \( 300 - 225 = 75 \)
The distance increases by 75 feet for every 1 minute increase in time.
Thus, the relationship can be described as a linear equation of the form:
\[ y = mx + b \]
Where:
- \( m \) is the slope (which is 75 in this case),
- \( b \) is the y-intercept (the value of y when x = 0).
Since all the points reveal consistent increases, we can calculate the y-intercept \( b \).
Using the first point (1, 75):
- Plug in the values to find \( b \):
\[ 75 = 75(1) + b \] \[ 75 = 75 + b \rightarrow b = 0 \]
Therefore, the equation that describes the relationship between time and distance is:
\[ y = 75x \]
This formula indicates that for every minute (x), the distance (y) increases by 75 feet.
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