The function ft 56 12cos0567t 689 represents the height o

The function f(t) = 56 + 12cos(0.567t - 6.89) represents the height of a wave (in miles) at t seconds. (a) What is the initial height of the wave? (b) What is the minimum height of the wave and at what item does it occur? (c) What is the maximum height of the wave and at what time does it occur? (d) At what time does the wave first reach 56 miles?

Solution

f(t) = 56 + 12cos(0.567t - 6.89)

a) Intial height : t =0

f(0) = 56 + 12cos(-0.689)

= 65.26

b) minium height : 56 -12 = 44

at point cos(0.567t - 6.89) = -1

0.567t - 6.89 = pi ; t = 17.69 sec

c) maximum height : 56+12 = 68

at point cos(0.567t - 6.89) = 1

t = 6.89/0.567 = 12.15 sec

d) f(t) = 56

56 = 56 +12cos(0.567t - 6.89)

0 = cos(0.567t - 6.89)

0.567t - 6.89 = pi/2

t = 14.92 sec