Question

question 4 (1 point) a pebble is embedded in the tread of a rotating bicycle wheel of diameter 60 cm. if the wheel rotates at 4 revolutions per second, determine a relationship between the height, h, of the pebble above the ground, in centimetres, as a function of time t, in seconds. a) ( h = 30 cosleft(\frac{1}{4}pi t
ight)+30 ) b) ( h = 30 cos(8pi t)+30 ) c) ( h = 60 cosleft(\frac{1}{4}pi t
ight) ) d) ( h = 60 cos(8pi t) ) question 5 (1 point) which graph of the following trigonometric functions has no zeros?

Explanation:

Step1: Determine the amplitude and vertical shift

The diameter of the wheel is \(d = 60\) cm, so the radius \(r=30\) cm. The vertical shift \(k\) (mid - line) is the height of the center of the wheel, \(k = 30\) cm. The amplitude \(A\) (maximum deviation from the mid - line) is equal to the radius of the wheel, \(A = 30\) cm.

Step2: Calculate the angular frequency \(\omega\)

The wheel rotates at \(f = 4\) revolutions per second. The formula for the angular frequency is \(\omega=2\pi f\). Substituting \(f = 4\) into the formula, we get \(\omega=2\pi\times4 = 8\pi\) radians per second.

Step3: Write the general form of the cosine function

The general form of a cosine function for vertical motion is \(h(t)=A\cos(\omega t)+k\). Substituting \(A = 30\), \(\omega=8\pi\), and \(k = 30\) into the formula, we get \(h(t)=30\cos(8\pi t)+30\). But if we assume the pebble starts at the top (a common assumption for such problems, and using the identity \(\cos(x)=\sin(x +\frac{\pi}{2})\)), and if we use the sine function form \(h(t)=A\sin(\omega t)+k\) (the phase - shift does not affect the form in terms of amplitude, period and vertical shift for our purpose of matching the options). The period \(T=\frac{2\pi}{\omega}\), and since \(f=\frac{1}{T}\), \(\omega = 8\pi\).

Let's check each option:

• Option a: For \(h = 30\sin(\frac{1}{4}\pi t)+30\), \(\omega=\frac{\pi}{4}\), \(T=\frac{2\pi}{\frac{\pi}{4}}=8\) seconds (not \(0.25\) seconds as \(f = 4\) implies \(T=\frac{1}{4}\) seconds)
• Option b: For \(h = 30\cos(8\pi t)+30\) (using the identity \(\cos(x)=\sin(x+\frac{\pi}{2})\), if we consider the sine form \(y = A\sin(\omega t)+k\) with \(A = 30\), \(k = 30\) and \(\omega=8\pi\) (because \(f = 4\), \(\omega=2\pi f\))

Answer:

B. \(h = 30\cos(8\pi t)+30\)