Expert: Bobby Soltani - 1/9/2005

Question
The tide at a boat dock can be modeled by the equation y=-2cos(pi/6t)+8, where t is the number of hours past noon and y is the height of the tide, in feet. For how many hours between t=0 and t=12 is the tide at least 7 feet?
I solved for cosine and got cos(pi/6 t)<=1/2, and then cos(30 degrees * t)<=1/2, but then I couldn\'t go further to solve for t.Please help.The answer is 8,but can\'t get that.Thank you

Answer
Hi Jeff,

You were pretty close to getting the answer.  You must recognize that cox (x degrees) = 1/2 when the angle is 60 or 300 degrees.  So, our equation becomes:
30t = 60
t = 2
and
30t = 300
t = 10
That means that the tide is at 7 feet at two hours an also at 10 hours past noon.  If we graph it on a calculator, or simply evaluate the expression at both sides of 2 and 10, we see that the curve (the tide) is above 7 feet from 2 to 10 and below seven feet the rest of the time.  Therefore, the tide is at least 7 feet for 8 hours.  Let me know if you have any questions.

Bobby