A particle moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ seconds, where $x(t)$ and $y(t)$ are measured in centimeters. It is known that $x'(t) = 8t - t^2$ and $y'(t) = -t + \sqrt{t^{1.2} + 20}$. At time $t = 2$ seconds, the particle is at the point $(3, 6)$. (a) Find the speed of the particle at time $t = 2$ seconds. Show the setup for your calculations. (b) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 2$. Show the setup for your calculations. (c) Find the $y$-coordinate of the position of the particle at the time $t = 0$. Show the setup for your calculations. (d) For $2 \leq t \leq 8$, the particle remains in the first quadrant. Find all times $t$ in the interval $2 \leq t \leq 8$ when the particle is moving toward the $x$-axis. Give a reason for your answer.
A particle moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ seconds, where $x(t)$ and $y(t)$ are measured in centimeters. It is known that $x'(t) = 8t - t^2$ and $y'(t) = -t + \sqrt{t^{1.2} + 20}$. At time $t = 2$ seconds, the particle is at the point $(3, 6)$.
(a) Find the speed of the particle at time $t = 2$ seconds. Show the setup for your calculations.
(b) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 2$. Show the setup for your calculations.
(c) Find the $y$-coordinate of the position of the particle at the time $t = 0$. Show the setup for your calculations.
(d) For $2 \leq t \leq 8$, the particle remains in the first quadrant. Find all times $t$ in the interval $2 \leq t \leq 8$ when the particle is moving toward the $x$-axis. Give a reason for your answer.