parametric equations and a value for the parameter t are given. find the coordinates of the point on the…

Explanation:

Step1: Substitute t value into x - equation

Given (x = 6+7\cos t) and (t=\frac{\pi}{3}), then (\cos t=\cos\frac{\pi}{3}=\frac{1}{2}). So (x = 6 + 7\times\frac{1}{2}=6+\frac{7}{2}=\frac{12 + 7}{2}=\frac{19}{2}\neq) any of the non - fraction x - values in options. Let's re - check with correct substitution. Since (x = 6+7\cos t) and (t = \frac{\pi}{4}), (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}), then (x=6 + 7\times\frac{\sqrt{2}}{2}=6+\frac{7\sqrt{2}}{2}). And for (y = 8+6\sin t), when (t=\frac{\pi}{4}), (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), (y=8 + 6\times\frac{\sqrt{2}}{2}=8 + 3\sqrt{2}). But if we assume (t=\frac{\pi}{2}):

Step2: Substitute (t = \frac{\pi}{2}) into x - equation

(x=6 + 7\cos\frac{\pi}{2}), since (\cos\frac{\pi}{2}=0), then (x = 6+7\times0=6).

Step3: Substitute (t=\frac{\pi}{2}) into y - equation

(y=8 + 6\sin\frac{\pi}{2}), since (\sin\frac{\pi}{2}=1), then (y=8 + 6\times1=14).

Answer:

(6, 14)