a curve in the plane is defined parametrically by the equations x = 2 cos(−t) and y = 5 sin(2t). find the…

Answer:

D. 5

Explanation:

Step1: Differentiate (x) with respect to (t)

Since (x = 2\cos(-t)=2\cos(t)), then (\frac{dx}{dt}=- 2\sin(t))

Step2: Differentiate (y) with respect to (t)

Since (y = 5\sin(2t)), using the chain - rule, (\frac{dy}{dt}=5\times2\cos(2t)=10\cos(2t))

Step3: Find (\frac{dy}{dx})

By the formula (\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}), we have (\frac{dy}{dx}=\frac{10\cos(2t)}{-2\sin(t)}=-\frac{5\cos(2t)}{\sin(t)})

Step4: Substitute (t = \frac{\pi}{6})

First, find (\cos(2t)) and (\sin(t)) when (t=\frac{\pi}{6}). (\cos(2\times\frac{\pi}{6})=\cos(\frac{\pi}{3})=\frac{1}{2}), (\sin(\frac{\pi}{6})=\frac{1}{2}) Then (\frac{dy}{dx}\big|_{t = \frac{\pi}{6}}=-\frac{5\times\frac{1}{2}}{\frac{1}{2}} = 5)