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 we know that (\cos\left(\frac{\pi}{2}\right)=0), so (h = 8).

Step2: Relate the height to the position of the minute - hand

On a clock, the minute - hand position is related to its height. The minute - hand makes a full circle in 60 minutes. The clock face is divided into 12 equal parts. Each part corresponds to an angle of (\frac{2\pi}{12}=\frac{\pi}{6}) radians or 5 minutes. We know that at (t = 0), the minute - hand position can be found by considering the cosine function's phase. The general form of a cosine function (y = A\cos(B(x - C))+D). Here (C = 15) means a phase shift. When (t = 0), the cosine function has a value corresponding to a position on the clock. Since the cosine function has a value of 0 at (\frac{\pi}{2}) and (-\frac{\pi}{2}), and we know that the minute - hand moves counter - clockwise. The minute - hand position at (t = 0) corresponds to 45 minutes (because of the phase shift in the cosine function). And 45 minutes on a clock corresponds to the 9 o'clock position.

Answer:

C. 9