The height of warer in a harbor rose to a max height of 15 f

The height of warer in a harbor rose to a max. height of 15 foot at 6 .00 p.m. and then dropped to a min level of 3 foot by 3.00 a.m. .the warwe level can be modedled by sine function write an equation hat represents the heigh of the water t hours after noon on yhe first day. Please explain completely how you find horizontal phase shift ?why?

Solution

maximum height = 15 feet

minimum height = 3 feet

amplitude = ( 15 - 3) /2 = 6

period = 2pi/b = 18

b = pi/ 9

1/4 of the period = 18/4 = 4.5

horizontal shift = amplitude - 4.5 = 6 - 4.5

= 1.5

vertical shift = maximum + minimum / 2 = ( 15 + 3) / 2

vertical shift = 9

hence, equation t hours after noon on first day is

y = 6 sin [ pi/9 ( t - 1.5 ) ] + 9