Math  /  Trigonometry

QuestionProblem 1. Professor Naehrig is running on a circular track in a uniform circular motion. Her position can be described through the parametric equations x=10+100cos(23πt+π4) and y=100sin(23πt+π4)x=10+100 \cos \left(\frac{2}{3} \pi t+\frac{\pi}{4}\right) \text { and } y=100 \sin \left(\frac{2}{3} \pi t+\frac{\pi}{4}\right) \text {, } where tt is measured in minutes, the angle is measured in radians, and lengths are measured in meters.
1. What are the circle center coordinates?
2. Explain the meaning of 23π\frac{2}{3} \pi in the parametric equations.
3. Explain the meaning of π4\frac{\pi}{4} in the parametric equations.
4. What is the radius of the circular track?
5. In coordinate system, sketch the circular track and Professor Naehrig's position at t=0,t=0.5,t=1.5,t=3t=0, t=0.5, t=1.5, t=3 minutes.
6. Find the first 3 times when Professor Naehrig's xx-coordinate is 503+1050 \sqrt{3}+10.
7. At the first time you found in the previous part, how far is she from the point (20,30)(20,30) ?
8. Find the first 3 times when Professor Naehrig's yy-coordinate is 50250 \sqrt{2}. What is her xx-position at those respective times?

Studdy Solution
To find the first 3 times when the yy-coordinate is 50250 \sqrt{2}, solve the equation:
100sin(23πt+π4)=502 100 \sin \left(\frac{2}{3} \pi t + \frac{\pi}{4}\right) = 50 \sqrt{2}
This simplifies to:
sin(23πt+π4)=22 \sin \left(\frac{2}{3} \pi t + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
Solve for tt to find the first 3 times, and then calculate the corresponding xx-positions using the parametric equation for xx.

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