Pre calc word problem

Question

The height in cm of the tip of the minute hand on a vertical clock face is a function, i (t), of the time, t, in minutes. The minute hand is 17 cm long, and the middle of the clock face is 219 cm above the ground. Find the midline and amplitude of the function i (t)

Answer 1

Taken in parts:

 

As the value of t increases the tip of the hand follows a cosine curve (Actually it is in harmonic motion). So basic function is 

f(t) = a(cos t - h) + k    (No phase shit)

 

The clock hand starts at (0, 219)

f(t) = a(cos t) + 219

 

The amplitude is the length of the hand, 17

f(t) = 17(cos t) + 219

Answer 2

Draw a picture!

 

On the hour (that is, minutes=0) the minute hand is at its maximum (that is, 219cm + 17 cm = 236 cm above the ground).  The minimum height of the tip of the minute hand occurs at 30 minutes past each hour (that is, 219 cm - 17 cm = 202 cm).  The midline for this oscillating function occurs a 15 minutes and again at 45 minutes past each hour (that is 219 cm -- the height of the middle of the clock face).

 

Now, realize that the problem statement does not require you to determine an equation for i(t) for the height of the tip of the minute hand.  However, if we were to plot the function, it would rise and fall between 236 and 202 with a midline at 219..  So, the "peak amplitude" (the measure from the midline to the maximum or to the minimum) is just 17 (the length of the minute hand).  [note: there are various amplitude definitions]