To create a formula relating time \( x \) (in minutes) to distance \( y \) (in feet), we can start by observing the data provided in the table:
\[ \begin{array}{|c|c|} \hline \text{Time (Minutes), } x & \text{Distance (Feet), } y \ \hline 1 & 75 \ 2 & 150 \ 3 & 225 \ 4 & 300 \ \hline \end{array} \]
From the data, you can see that as the time increases by 1 minute, the distance increases by 75 feet. This indicates a linear relationship between \( x \) and \( y \).
To derive the linear equation, we can use the slope-intercept form of a line, which is expressed as:
\[ y = mx + b \]
Where \( m \) represents the slope and \( b \) is the y-intercept.
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Finding the slope \( m \): \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{150 - 75}{2 - 1} = \frac{75}{1} = 75 \]
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Finding the y-intercept \( b \): When \( x = 0 \), \( y \) can be calculated by observing the relation: \[ y = mx + b \implies y = 75(0) + b \implies y = b \] However, since there is no data at \( x = 0 \), we can simply use one of the points. Let's use \( (1, 75) \) to find \( b \): \[ 75 = 75(1) + b \implies 75 = 75 + b \implies b = 0 \]
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Final Equation: Putting it all together, the formula that relates \( x \) and \( y \) is: \[ y = 75x \]
Thus, the formula that represents the relationship between time in minutes \( x \) and distance in feet \( y \) is:
\[ y = 75x \]
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