Lorelei uses a large oval-shaped tank to feed her horses. Due to recent thunderstorms the tank is completely full of dirty water that needs to be removed. At time t = 0, Lorelei starts to drain the tank at a rate modeled by g(t), measured in gallons per hour, where g is given by the piecewise function: (t-2)(3t+2)/t-2 0 < t < 2
g(t) = k 2 < t < 4
ae^0.49t 4 < t < 5
a. What values of k and a make g(x) continuous at both t = 2 and t = 4?
b. The average acceleration of the draining water from time a to time a + h is given by:
average acceleration = g(a + h) - g(a)/h
Is the average acceleration of g(t) higher from t = 0 to t = 1 or from t = 4 to t = 5?
c. The cylindrical horse feeder in the neighboring pasture is also full of water. At time t = 0, Lorelei's friend starts to drain the water from this feeder at a rate modeled by s(t).
s (t) = 3t
Let h(t) = g(t) - s(t). Apply the Intermediate Value Theorem to the function h on the interval 0 < t < 5 to justify that there exists a time t, 0 < t < 5, at which the heights of water in the two feeders are changing at the same rate.