To create a linear function in the form \( y = mx + b \) that represents the distance \( d \) the car has traveled in relation to time \( t \), we need to determine the slope \( m \) and the y-intercept \( b \).
From the problem, we know that:
- After \( t = 0 \) seconds, the distance \( d = 0 \) feet (since the car has just passed the timing device).
- After \( t = 8 \) seconds, the distance \( d = 840 \) feet.
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Finding the slope \( m \): The slope \( m \) represents the speed of the car in feet per second and can be calculated as:
\[ m = \frac{\Delta d}{\Delta t} = \frac{840 \text{ feet} - 0 \text{ feet}}{8 \text{ seconds} - 0 \text{ seconds}} = \frac{840}{8} = 105 \text{ feet per second} \]
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Finding the y-intercept \( b \): Since at time \( t = 0 \) the distance \( d = 0 \), the y-intercept \( b \) is:
\[ b = 0 \]
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Writing the linear function: Now, we can write the linear function \( d(t) \) in the form \( d = mt + b \):
\[ d(t) = 105t + 0 \]
Thus, the linear function that represents the distance \( d \) the car has traveled after \( t \) seconds is:
\[ \boxed{d(t) = 105t} \]