To determine how long the distance Rachel has to drive, we need to evaluate the distance when [tex]\( t = 0 \)[/tex].
The formula representing the distance [tex]\( D \)[/tex] Rachel has left to drive after [tex]\( t \)[/tex] seconds is given by:
[tex]\[ D = -38t + 220 \][/tex]
1. Initial Distance:
- When [tex]\( t = 0 \)[/tex], Rachel has just started to drive. Plugging [tex]\( t = 0 \)[/tex] into the equation:
[tex]\[ D = -38(0) + 220 \][/tex]
[tex]\[ D = 220 \][/tex]
This means that the initial distance Rachel has to drive is 220 meters.
Next, we need to determine the time it takes for Rachel to escape, which happens when the remaining distance [tex]\( D \)[/tex] is 0.
2. Finding the Time to Escape:
- We set [tex]\( D = 0 \)[/tex] and solve for [tex]\( t \)[/tex]:
[tex]\[ 0 = -38t + 220 \][/tex]
[tex]\[ 38t = 220 \][/tex]
[tex]\[ t = \frac{220}{38} \][/tex]
When calculating [tex]\( t = \frac{220}{38} \)[/tex], we get approximately:
[tex]\[ t \approx 5.789 \text{ seconds} \][/tex]
Thus, it takes Rachel approximately 5.789 seconds to escape the building.
To summarize:
- The initial distance Rachel has to drive is 220 meters.
- The time it takes for Rachel to escape the building is approximately 5.789 seconds.
