Trigonometry Word Problem

Question

Suppose the depth of the tide in a certain harbor can be modeled by y=23 + 4 cos (pi/6 x t), where y is the water depth in feet and t is the time in hours. Consider a day in which t=0 represents 12:00 midnight. For that day, when are high tide and low tide and what is the depth of each?

 

a) high tide at 6:00 am and 6:00 pm, depth 23 ft 

low tide at 12:00 noon and 12:00 midnight, depth 4 ft

b) high tide at 12:00 noon, depth 23 ft
low tide at 6:00 am, depth 4 ft

c) high tide at 6:00 am and 6:00 pm, depth 27 ft
low tide at 12:00 noon and 12:00 midnight, depth 19 ft

d) high tide at 12:00 midnight and 12:00 non,depth 27 ft
low tide at 6:00 am and 6:00 pm, depth 19 ft

 

I want to understand hoe to get to the answer, please explain the procedure, it be grateful!!

 

Answer 1

High tide will occur when the argument of the cosine function is 0, because cos 0 = 1 which is its maximum value,

 

so midnight is a high tide.

 

 

The maximum value for the cosine function is 1,

so the high tide will be  23 + 4(1) = 27 ft at high tide

 

The minimum value of the cosine function is -1,

so low tide will be 23 + 4(-1) or 19 ft.

 

Further calculations are possible, but let us save them to confirm the only answer possible, c.

 

 

Does low tide, in fact, occur at 6 am?  

 

19 = 23 + 4 cos [(1/6)π6] =

 

23 + 4 cos π =

 

23 + 4(-1) = 23 - 4 = 19