A runner is jogging around a level circular track. His distance north
of thecentre of the track in meters is given by $60\cos0.08t$, where t is
measured in seconds.a) How long does it take the runner to complete one lap?
b) What is the length of the track?
c) At what speed is the runner jogging?
I'm having trouble understanding this question, because the equation is notated as shown above with no parentheses, therefore not allowing me to comprehend whether it's $60\cos(0.08)t$ or $60\cos(0.08t)$. However, the answer book says that the answers are 78.5s, 377m and 4.8m/s.
What I tried: I assumed because the amplitude of the wave is 60 that the radius of the circle would be 60 and the circumference would be $\approx$376.99. After that I checked my calculator to see what time that would be (by inputting the circumference as one horizontal line and the equation as another line and checking the intersection point), but the answer I got was $\approx$6.2..? Obviously a runner can't run 377 meters in 6 seconds so I'm not sure what went wrong here.
I would like to be able to do all 3 problems, (a), (b), and (c), however I understand there is a policy against people doing the work for me so if someone could explain to me what I've done wrong I would really appreciate it, as well as how to fix it.
Best Answer
Regarding the notation, because the position of the runner varies with time, $t$ needs to be a parameter of the equation. So you should be able to safely interpret the given function as $d(t)=60\cdot\cos(0.08t)$.
I think you forgot to adjust your period for the 0.08 factor. So, $b=0.08=\frac{2}{25}$, making the period $\frac{2\pi}{b}=\frac{2\pi}{\frac{2}{25}}=25\pi$. Thus the runner will circle the track in $25\pi$ seconds.
You correctly calculated the circumference of the track, so now that you know how long it takes to make one revolution, you should be able to divide to find the runner's speed.
Hope this helps!