Question

The height, h, in feet of the tip of the hour hand of a wall clock varies from 9 feet to 10 feet. Which of the following equations can be used to model the height as a function of time, t, in hours? Assume that the time at t = 0 is 12:00 a.m

Answer 1

y=Acos(p)+m, A=amplitude, p=period, m=midline, in this case:

A=1/2, p=360(t/12)=30t, m=(10-9)/2+9=9.5 so

h(t)=(1/2)cos(30t)+9.5

Answer 2

The required equation is
`y=0.5\cos ((\pi t)/(6))+9.5`.

Explanation:

It is given that the height, h, in feet of the tip of the hour hand of a wall clock varies from 9 feet to 10 feet. It means the minimum height is 9 feet and the maximum height is 10.

At x=0 is 12:00 a.m, it means the function is maximum at x=0. So, the general equation is defined as

`y=a\cos(\omega t)+b`

Where, a is amplitude, w is period, t is time in hours and b is midline.

Midline is the average of minimum and maximum height.

`b=(9+10)/(2)=9.5`

Amplitude is the distance of maximum of minimum value from midline.

`a=10-9.5=0.5`

Now, the equation can be written as

`y=0.5\cos(wt)+9.5` .... (1)

At t = 0 is 12:00 a.m, so t=3 is 3:00 a.m and the y=9.5 at x=3.

`9.5=0.5\cos(3w)+9.5`

`\cos(3w)=0`

`\cos(3w)=\cos((\pi)/(2))`

`3w=(\pi)/(2)`

`w=(\pi)/(6)`

Substitute this value in equation (1).

`y=0.5\cos ((\pi t)/(6))+9.5`

Therefore the required equation is
`y=0.5\cos ((\pi t)/(6))+9.5`.