On a cross-country journey the distance travelled at a given time by a wagon caravan is given by the equation:
p(t) = t^5 + 2t^3 - t^2 + 12
Starting at t = -1, one of the wagons starts moving on its own at a constant rate, and rejoins the caravan at t = 1. What is the equation that this lone wagon follows during this time interval?
To find the equation that the lone wagon follows during the time interval from t = -1 to t = 1, we need to subtract the equation of the caravan from the total distance equation.
Given the equation for the total distance travelled by the caravan:
p(t) = t^5 + 2t^3 - t^2 + 12
We can find the equation for the lone wagon by subtracting the equation for the caravan. Since the lone wagon starts at t = -1 and rejoins the caravan at t = 1, we need to evaluate the total distance equation at those two points:
p(-1) = (-1)^5 + 2(-1)^3 - (-1)^2 + 12
= -1 + 2(-1) - 1 + 12
= -1 - 2 - 1 + 12
= 8
p(1) = (1)^5 + 2(1)^3 - (1)^2 + 12
= 1 + 2(1) - 1 + 12
= 1 + 2 - 1 + 12
= 14
Now, we can subtract the distances travelled by the caravan at t = -1 and t = 1 from the total distance equation:
lone_wagon(t) = p(t) - p(-1) - [p(1) - p(-1)]
Let's simplify this equation:
lone_wagon(t) = t^5 + 2t^3 - t^2 + 12 - 8 - (14 - 8)
= t^5 + 2t^3 - t^2 + 12 - 8 - 6
= t^5 + 2t^3 - t^2 - 2
Therefore, the equation that the lone wagon follows during the time interval from t = -1 to t = 1 is:
lone_wagon(t) = t^5 + 2t^3 - t^2 - 2