The parametric equations for the paths of two projectiles are given below. At what rate is the distance between the two objects changing at $t = \frac { \pi } { 2 } ? \text { Round your answer to two decimal }$ places. $\begin{array} { l }
x _ { 1 } = 12 \cos 2 t , y _ { 1 } = 6 \sin 2 t \
x _ { 2 } = 6 \cos t , y _ { 2 } = 7 \sin t
\end{array}$
A) $0.86$
B) $1.54$
C) $0.58$
D) $- 1.73$
E) $- 0.65$

Question

The parametric equations for the paths of two projectiles are given below. At what rate is the distance between the two objects changing at $t = \frac { \pi } { 2 } ? \text { Round your answer to two decimal }$ places. $\begin{array} { l } 
x _ { 1 } = 12 \cos 2 t , y _ { 1 } = 6 \sin 2 t \
x _ { 2 } = 6 \cos t , y _ { 2 } = 7 \sin t
\end{array}$
A) $0.86$
B) $1.54$
C) $0.58$
D) $- 1.73$
E) $- 0.65$

Quizplus · Accepted Answer

The correct answer is A because the rate of change of the distance between the two objects is given by the derivative of the distance function with respect to time. The distance function is given by $d(t) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$, so the derivative of the distance function is given by $$\frac{d}{dt}d(t) = \frac{1}{2\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}}\left(2(x_1 - x_2)\frac{dx_1}{dt} + 2(y_1 - y_2)\frac{dy_1}{dt}\right)$$At $t = \frac{\pi}{2}$, we have $$x_1 = 12 \cos 2\left(\frac{\pi}{2}\right) = 0, \quad y_1 = 6 \sin 2\left(\frac{\pi}{2}\right) = 6,$$$$x_2 = 6 \cos \left(\frac{\pi}{2}\right) = 0, \quad y_2 = 7 \sin \left(\frac{\pi}{2}\right) = 7.$$So, $$\frac{d}{dt}d(t) = \frac{1}{2\sqrt{(0 - 0)^2 + (6 - 7)^2}}\left(2(0 - 0)\frac{d}{dt}(0) + 2(6 - 7)\frac{d}{dt}(6)\right) = \frac{-2}{2} = -1.$$Therefore, the rate of change of the distance between the two objects at $t = \frac{\pi}{2}$ is $-1$.

The Parametric Equations for the Paths of Two Projectiles Are t=π2? Round your answer to two decimal t = \frac { \pi } { 2 } ? \text { Round your answer to two decimal }t=2π​? Round your answer to two decimal

The Parametric Equations for the Paths of Two Projectiles Are $t = \frac { \pi } { 2 } ? \text { Round your answer to two decimal }$