The position of a roller coaster car at time $t$ seconds can be modeled parametrically by $$\begin{aligned} & x ( t ) = 10 t + 4 \sin t \\ & y ( t ) = ( 20 - t ) ( 1 - \cos t ) , \end{aligned}$$ where $x$ and $y$ are measured in meters, over the time interval $0 \leq t \leq 18$ seconds. The derivatives of these functions are given by $$\begin{aligned} & x ^ { \prime } ( t ) = 10 + 4 \cos t \\ & y ^ { \prime } ( t ) = ( 20 - t ) \sin t + \cos t - 1 \end{aligned}$$
(a) Find the slope of the path at time $t = 2$. Show the computations that lead to your answer.
(b) Find the acceleration vector of the car at the time when the car's horizontal position is $x = 140$.
(c) Find the time $t$ at which the car is at its maximum height, and find the speed, in $\mathrm { m } / \mathrm { sec }$, of the car at this time.
(d) For $0 < t < 18$, there are two times at which the car is at ground level ( $y = 0$ ). Find these two times and write an expression that gives the average speed, in $\mathrm { m } / \mathrm { sec }$, of the car between these two times. Do not evaluate the expression.