word problem . Pls help

Question

The tide at a given bay can be modelled by an equation of the form d = A cos ( B(t+C)
)+D, where d represents the depth of water at the bay and t represents the number of hours from 12:00 am on the 23^rd of August. At 4:00 am on the 23^rd of August, low tide was recorded at a depth of 6.7 metres. The high tide occurred at 11:00 am at a depth of 10.97 meters.

(a) Sketch a graph of the tide and label clearly the time (i.e. xx:xx am/pm, date) when the next high tide occurred after 11:00 am on the 23^rd of August.

 

(b) Write the trigonometric function to model the tide (depth of the water) at the bay.

 

(c) A passenger ship cannot be in the bay when the depth of water is less than or equal to 7.62 metres. Write the time interval/s when a passenger ship is allowed to dock in the bay on the 1^st of September.

Answer 1

Hi Rikah.

This is an amazing problem...  

 

First. The max value is at high tide =10.97

The min value is at low tide = 6.7

 

Average these and you will get the center of the cosine wave

(10.97+6.7)/2 = 8.835 meters.

This is the height "usually" and represents a vertical shift of "up" 8.835.  This is the value of "D" in the equation. d=Acos(B(t+C))+8.835

 

"A" is how far up and how far down the max and min are from this baseline (average).  So A = 10.97 - 8.835 = 2.135.  To check, should get the same thing when 8.835 - 6.7 = 2.135

So 2.135 is the "amplitude" of the cosine wave and now have

d = 2.135 cos(B(t+C)) + 8.835

 

At low tide, need cos(B(t+C)) to be at its minimum, -1

So for  cos(anything) = -1, the (anything) must = pi.

B(t+C) = pi.  Low tide is when t=4.

 

So B(4+C)= pi. 

 

At high tide, need cos(B(t+C)) to be at its maximum, +1

cos(anything) = 1 means (anything) =0.

So B(t+C) = 0.  At high tide t=11 so

B(11+C)=0. This means that either B=0 or that 11+C=0

B cannot = 0 or the previous equation B(4+C)=pi would result in 0=pi which is not true.  So C must = -11.

 

Now we have:

d=2.135 cos(B(t - 11) + 8.835

 

Last need to determine B from B(4 + (-11)) = pi

-7 B = pi

B = pi/(-7)

 

Finally...

d = 2.135 cos((-pi/7)(t - 11) + 8.835

I am out of time...  Can get rest of info and graph it from here.  Must represent Sept 1st as the number of hours from 12:00 am on Aug 23.