To create a formula from the given data points, we need to express the relationship between time (x) and distance (y). From the provided values:
- When \(x = 1\), \(y = 75\)
- When \(x = 2\), \(y = 150\)
- When \(x = 3\), \(y = 225\)
- When \(x = 4\), \(y = 300\)
We can observe that for every increase of 1 minute in time, the distance increases by 75 feet. This indicates a linear relationship between time and distance.
To find the linear equation in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:
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Calculate the slope (m): The change in distance (y) over the change in time (x) can be calculated using any two points. Using the first two points: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{150 - 75}{2 - 1} = \frac{75}{1} = 75 \]
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Use one point to find the y-intercept (b): We can use the point (1, 75): \[ y = mx + b \implies 75 = 75(1) + b \implies 75 = 75 + b \implies b = 0 \]
Thus, the linear equation that expresses the relationship between time and distance is: \[ y = 75x \]
This formula indicates that for every minute, the distance increases by 75 feet, with no initial distance when time is zero.
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