the height, h, in feet of the tip of the minute hand of a wall clock as a function of time, t, in minutes…

the height, h, in feet of the tip of the minute hand of a wall clock as a function of time, t, in minutes can be modeled by the equation $h = 0.75cosleft(\frac{pi}{30}(t - 15)\right)+8$. which number (from 1 to 12) is the minute hand pointing to at $t = 0$?\n3\n6\n9\n12

the height, h, in feet of the tip of the minute hand of a wall clock as a function of time, t, in minutes can be modeled by the equation $h = 0.75cosleft(\frac{pi}{30}(t - 15)\right)+8$. which number (from 1 to 12) is the minute hand pointing to at $t = 0$?\n3\n6\n9\n12

Answer

Explanation:

Step1: Substitute t = 0 into the equation

Substitute (t = 0) into (h=0.75\cos\left(\frac{\pi}{30}(t - 15)\right)+8). We get (h = 0.75\cos\left(\frac{\pi}{30}(0 - 15)\right)+8=0.75\cos\left(-\frac{\pi}{2}\right)+8). Since (\cos(-\alpha)=\cos(\alpha)), then (h = 0.75\cos\left(\frac{\pi}{2}\right)+8). And (\cos\left(\frac{\pi}{2}\right)=0), so (h = 8).

Step2: Analyze the position of the minute - hand

The minute - hand of a clock makes a full - circle (360 degrees or (2\pi) radians) in 60 minutes. The general form of a cosine function for the height of the minute - hand can be related to its angular position. We know that when (t = 0), the argument of the cosine function is (\frac{\pi}{30}(0 - 15)=-\frac{\pi}{2}). In terms of the clock face, a full - circle is divided into 12 equal parts (hours). Each part corresponds to an angle of (\frac{2\pi}{12}=\frac{\pi}{6}) radians. If we consider the starting position of the minute - hand at (t = 0), an angle of (-\frac{\pi}{2}) radians from the 12 - o'clock position (since the cosine function is centered around the vertical position of the minute - hand) corresponds to the 9 - o'clock position.

Answer:

C. 9