Question

7. High tide in Lake Ontario is 5.55 m high and occurs at 1 am. Low tide is 1.00 m high and occurs at 1 pm.. Time is in hours from midnight and height is in meters. a. b. c. Derive a cosine equation to represent the height of the wave at any given time of the day How high are the waves at 2:55 pm? What are the first two times that the waves have a height of 4.25 m?

Answer 1

a) Average-Value = 1/2 * (High-Val + Low-Val) = 1/2 * (5.55+1.00) = 3.275

The Maxima occurs at 1 am and the minima occurs at 1 pm, which means in the 12 hour gap, the cos function value will change from 0 to pi.

Hence for each hour, the value must be pi/12

r(t) = 3.275 + 2.275cos-(t-1)) 12

t-1 must be present, because the maxima is coming at t=1 am, which represent 1 hour and maxima is coming at 13 hours, which is 1 pm

Hence the equation is justifying all the values

b)

At 2:55 pm, the value of t will be 14 + 55/60 (since 1 hour contains 60 minutes)

2(2.55 pm) = 3.275+2.275cos_(14t_-1)) = 3.275+2.275(-0.8767) 12 60 1.2805 m

Hence the height of the tide will be 1.2805m

c)

4.25-3.275+ 2.275cos-(t-1)) 12

-(t-1)) = 0.4285 12 cos( COS

t-1-4.309

5.309

Hence the first time will be 5 am + 0.309 * 60 = 5 hr 18.54 mins am

The next time will be 2*pi - 4.309*pi/12, which will be 1.6408

(t-1) = 12 * 1.6408

t = 20.68

which represents the time of 8 hours 40 min pm