# What am I doing wrong in this trigonometry/rate simulation problem?

I'm refreshing on some trig and cannot figure out how to solve this non-realistic word problem simulating a person walking in a circle.

A person is located at the point (8,0) at time, t = 0, and walks at the rate of 8 yards/min in a counterclockwise direction along the circle centered at the origin of radius 8.

1. What are the coordinates (x,y) which give the location of the person after 29 minutes?

2. After how many minutes will the person return to their starting location (8,0)?

I know that I'm supposed to be using the unit circle, but can't seem to come up with an answer. Here's what I've done:

i.) The circumference of the circle is 16Pi?

ii.) The distance traveled in 29 min is 232 yards?

iii.) Hypotenuse is 232?

I know that I somehow have to make a triangle and use trig functions to answer the 2 questions but I don't understand what the side lengths would be. Starting out, I think the adjacent side is 8 yards? And I believe x = r*cos(z) and y = r*sin(z)?

I just need some help getting started. Anything will be greatly appreciated. Thanks!

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The hypotenuse is $8$. Calculate the number of revolutions travelled. It is $\frac{232}{16\pi}\approx 4.6155$. That's $4$ complete revolutions plus about $221.5776$ degrees. Call this angle $\theta$. Your coordinates will be $x=r\cos\theta$, $y=r\sin\theta$, where $r=8$. Do check my arithmetic! Depending on your course, you may want to work in radiams rather than degrees. – André Nicolas Sep 5 '13 at 23:31
@AndréNicolas so theta would = 232/16pi? Or would it be 221.5776 degrees (converted to radians)? – Cozen Sep 5 '13 at 23:42
That ratio is the number of revolutions. I removed the leading $4$, but you actually don't need to. Then multiply by $360$ to get the angle in degrees. Or multiply by $2\pi$ to get the angle in radians. That's actually nicer in this problem, since the $\pi$ cancel, and we get $\frac{232}{8}=29$. No accident of course. I used degrees because I guessed you were working in degrees. – André Nicolas Sep 5 '13 at 23:50
@AndréNicolas Perfect, thanks! The last thing I can't figure out is how long it will take before the person gets back to (8,0) – Cozen Sep 6 '13 at 4:54
You mean how long after the $29$ minutes? Let's find the time for $5$ revolutions, and take away $29$. Five rev. is $80\pi$, at speed $8$ takes $10\pi$. Subtract the $29$ already travelled. – André Nicolas Sep 6 '13 at 5:10

We can work in degrees or in radians. The advantage of degrees is that our intuition is better developed. We will set up the computations in both.

The circumference of the circle is $16\pi$. If we have travelled for $29$ minutes at $8$ yards per minute, we have travelled a distance $(29)(8)$.

Divide by $16\pi$. That gives us the number of revolutions. So we have travelled through $\frac{(29)(8)}{16\pi}$ revolutions. Thus we have travelled through an angle of $$\frac{(29)(8)}{16\pi}\cdot 360$$ degrees.

In radians, the angle we have travelled through looks much simpler. There are $2\pi$ radians per revolution, so we have travelled through $$\frac{(29)(8)}{16\pi}(2\pi)=29$$ radians. The reason the answer is so simple is that at $8$ yards per minute, which matches the radius, we are rotating at $1$ radian per minute.

Whatever measure we use, if $\theta$ is the angle we have travelled through, our location $(x,y)$ is given by $$x=8\cos\theta,\qquad y=8\sin\theta.$$ Now we can compute. Put the calculator in radian mode, and find $8\cos(29)$ and $8\sin(29)$.

The question about return to the starting point is a little ambiguous. The amount of time for a complete revolution is the circumference divided by the speed, so this is $\frac{16\pi}{8}$ minutes.

But if we have been travelling for $29$ minutes, then we have gone through $\frac{(29)(8)}{16\pi}$ revolutions, somewhat more than $4.6$. To get to the start point, we need to complete the fifth revolution.

Doing $5$ complete revolutions take time $\frac{(5)(16\pi)}{8}=10\pi$ minutes. So if we have already travelled for $29$ minutes, the additional time we need to return to $(8,0)$ is $10\pi-29$ minutes.

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