Question

9. A car moving at a constant speed passes a timing
device at t = 0. After 8 seconds, the car has
traveled 840 feet. What linear function in the
form y = mx + b represents the distance in feet,
d, the car has traveled any number of seconds, t, after passing the timing device?

Answer 1

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.

1. **Finding the slope \( m \):**
The slope \( m \) represents the speed of the car in feet per second and can be calculated as:

\[
m = \frac{\Delta d}{\Delta t} = \frac{840 \text{ feet} - 0 \text{ feet}}{8 \text{ seconds} - 0 \text{ seconds}} = \frac{840}{8} = 105 \text{ feet per second}
\]

2. **Finding the y-intercept \( b \):**
Since at time \( t = 0 \) the distance \( d = 0 \), the y-intercept \( b \) is:

\[
b = 0
\]

3. **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:

\[
\boxed{d(t) = 105t}
\]