On a summer day in Cape Cod, the depth of the water at a dock was 4ft at low tide at 2:00 AM. At high tide 5 hours later, the heist of the water at the dock rose to 14 feet. Write a cosine function to model the height of the water at the dock x hours after the day began at midnight?
+128475 We will have an equation of this form
Height = A cos ( Bx + C) + D
A = the amplitude = [ 14 - 4 ] / 2 = 5
The period = 10 hrs....so 10 = 2pi / B ⇒ B = 2pi/10 = pi/5
D is the midline = [ 14 + 4 ] /2 = 18 /2 = 9
C is the phase shift and is a little tough to figure
At 2AM, x = 2, the height is 4 ft.....so we need to solve this
4 = 5 cos ( pi/5 * 2 + C ) + 9 subtract 9 from both sides
-5 = 5 cos ( 2pi/5 + C) divide both sides by 5
-1 = cos (2pi/5 + C) we need to take the arccos
arccos (-1) = 2pi/5 +C
pi = 2pi/5 + C
C = pi - 2pi/5 = [ 5pi -2pi] / 5 = 3pi/5
So....the function is
Height = 5cos [ ( pi /5) x + 3pi/5 ) + 9
Here's the graph : https://www.desmos.com/calculator/igberusuwx
