SOLUTION: The water at a local beach has an average depth of one meter at low tide and an average depth at high tide of 8m. One cycle of the tides takes approximately 12 hours. This periodic

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Question 1183639: The water at a local beach has an average depth of one meter at low tide and an average depth at high tide of 8m. One cycle of the tides takes approximately 12 hours. This periodic motion can be modelled by the function
d(t)=3.5 cos [(pi/6)t]+4.5 where d(t) represents the depth of the water, in metres, at a time t, in hours. This equation assumes that the water level is at high tide at time zero (midnight). Many people dive into the water from a nearby dock. If the water must be at least 3m deep to dive safely, when during the first 24 hour period (in hours) would it be safe to dive off the dock? Please give your answer in all ranges of hours possible.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The periodic motion is modelled by
d%28t%29+=+3.5%2Acos%28pi%2F6%29%2At+%2B+4.5, where t = 0 starts at midnight.
We want to find the values of t where d%28t%29+=+3.5%2Acos%28pi%2F6%29%2At+%2B+4.5+%3E=+3.

===> 3.5%2Acos%28pi%2F6%29%2At+%2B+4.5+%3E=+3+

===> 3.5%2Acos%28pi%2F6%29%2At+=+-1.5 ===> cos%28pi%2F6%29%2At+=+-3%2F7

===>+t+=+%286%2Fpi%29%2Acos%5E-1%28-3%2F7%29 ~ 3.85 hours after midnight.

The ranges of time where it is safe to dive, i.e., d%28t%29+%3E=+3, are then
[0, 3.85], [8.15, 15.85], and [21.85, 24].

These correspond to the following times when it is safe to dive:
12:00 am - 3.51 am, 8:09 am - 3:51 pm, and 9:51 pm - 12:00 am.