A truck traveling at 16 m/s is 24 meters behind the bus at t = 0. The equation d = -24 + 16t gives the position of the truck. At what time will the truck be the same distance from the intersection as the bus? What will that distance be?
To find the time at which the truck will be the same distance from the intersection as the bus, we need to set the distances traveled by both the truck and the bus equal to each other.
First, let's set up an equation for the position of the bus. Let's assume the position of the bus can be given by the equation d = bt, where d represents the distance traveled by the bus and b represents the bus's speed.
From the given information, we know that the truck is initially 24 meters behind the bus at t = 0, so the position of the truck can be represented as d = -24 + 16t.
To find the time at which the truck will be the same distance from the intersection as the bus, we need to equate the two distances traveled and solve for t:
bt = -24 + 16t
Rearranging the equation, we get:
bt - 16t = -24
Factoring out the t, we get:
t(b - 16) = -24
Dividing both sides by (b - 16), we get:
t = -24 / (b - 16)
To find the distance at this time, we can substitute the value of t into either the equation for the truck or the bus.
Let's substitute t = -24 / (b - 16) into the equation for the truck:
d = -24 + 16(-24 / (b - 16))
d = -24 - 384 / (b - 16)
Simplifying the equation further, we get:
d = (-24(b - 16) - 384) / (b - 16)
d = (-24b + 384 - 384) / (b - 16)
d = -24b / (b - 16)
Therefore, at time t = -24 / (b - 16), the distance of the truck from the intersection is given by -24b / (b - 16).