A train driver is driving along and he suddenly puts on the brakes. After he holds the brakes for t seconds, the train's sliding distance can be measured as d(t) = 30t - 0.6t^2 (in meters). When the train stops, how far did the train slide (in meters) after the train driver put on the brakes?
To find the distance the train slides after the train driver put on the brakes, we need to determine the value of "d(t)" when the train stops. The train stops when its velocity becomes zero, so we need to solve the equation for t when d(t) = 0.
The equation for the distance function is d(t) = 30t - 0.6t^2. Setting d(t) to zero and solving for t, we have:
0 = 30t - 0.6t^2
Rearranging the equation:
0.6t^2 - 30t = 0
Dividing both sides by 0.6:
t^2 - 50t = 0
Now, factoring out a t:
t(t - 50) = 0
From the equation above, we have two possible solutions:
1. t = 0 (implies the train didn't move, which is not the case here)
2. t - 50 = 0, which gives t = 50
Therefore, the train stops after 50 seconds.
Now, substituting t = 50 into the equation for d(t):
d(t) = 30t - 0.6t^2
= 30(50) - 0.6(50)^2
= 1500 - 0.6(2500)
= 1500 - 1500
= 0
So, the train slides a distance of 0 meters after the train driver put on the brakes and the train comes to a stop.