91Ó°ÊÓ

The diagram shows a merry-go-round that is turning counterclockwise at a constant rate, making 2 revolutions in 1 minute. On the merry-go-round are horses \(A, B, C,\) and \(D\) at 4 meters from the center and horses \(E, F,\) and \(G\) at 8 meters from the center. There is a function \(a(t)\) that gives the distance the horse \(A\) is from the \(y\) -axis (this is the \(x\) -coordinate of the position \(A\) is in ) as a function of time \(t\) (measured in minutes). Similarly, \(b(t)\) gives the \(x\) -coordinate for \(B\) as a function of time, and so on. Assume that the diagram shows the situation at time \(t=0\). (Check your book to see figure) (a) Which of the following functions does \(a(t)\) equal? $$\begin{array}{ll}4 \cos t, & 4 \cos \pi t, \quad 4 \cos 2 t, \quad 4 \cos 2 \pi t \\\4 \cos \left(\frac{1}{2} t\right), & 4 \cos ((\pi / 2) t), \quad 4 \cos 4 \pi t\end{array}$$ Explain. (b) Describe the functions \(b(t), c(t), d(t),\) and so on using the cosine function: $$\begin{array}{l}b(t)=\longrightarrow(t)=\longrightarrow d(t)= \\\e(t)=\longrightarrow f(t)=\longrightarrow g(t)=\end{array}$$ (c) Suppose the \(x\) -coordinate of a horse \(S\) is given by the function \(4 \cos (4 \pi t-(5 \pi / 6))\) and the \(x\) -coordinate of another horse \(T\) is given by \(8 \cos (4 \pi t-(\pi / 3))\) Where are these horses located in relation to the rest of the horses? Mark the positions of \(T\) and \(S\) at \(t=0\) into the figure.

The percentage of the face of the moon that is illuminated (as seen from earth) on day \(t\) of the lunar month is given by $$g(t)=.5\left(1-\cos \frac{2 \pi t}{29.5}\right)$$ (a) What percentage of the face of the moon is illuminated on day 0? Day 10? Day 22? (b) Construct appropriate tables to confirm that \(g\) is a periodic function with period 29.5 days. (c) When does a full moon occur \((g(t)=1) ?\)

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 4.25 minutes, \(5 \mathrm{rpm}\)

Here is proof that the sine function has period \(2 \pi .\) We saw in the text that \(\sin (t+2 \pi)=\sin t\) for every \(t .\) We must show that there is no positive number smaller than \(2 \pi\) with this property. Do this as follows: (a) Find a number \(t\) such that \(\sin (t+\pi) \neq \sin t\) (b) Find all numbers \(k\) such that \(0