Tides the table at the right shows the times for high tide and low tide of one day. The markings on the side of the local pier showed a high tide of 7ft and a low tide of 4ft on the previous day.
tide chart
High Tide 4:03am
Low Tide 10:14am
High Tide 4:25pm
Low Tide 10:36pm
a.What is the average depth of water at the pier?What is the amplitude of the variation from the average depth?
-use the cosine function:y=acosbt
The amplitude is |a|=1/2 (maximum height-minimum height)
=1/2 (7-4)=1.5 & since we start with high tide
then a >0 so a=1.5ft
Average depth =((maximum high tide+minimum low tide))/2=(7+4)/2=11/2=5.5 ft
b.How long is one cycle of the tide?
I found my a part but i dont know how to find b? equation use for part b is period =2π/B
the answer is 12hr 22min or B= 60pi/371 i dont understand how they got the answer to part b
you know that the distance between a max and a min for the cosine function is 1/2 period. The table shows some variation in that, but it is clear that there are two high and two low tides each day. So the period will be about 12 hours, right?
More like 12.5 hours, about an hour later every day. (In Gloucester at least :)
To find the value of b in the equation y = acos(bt) for part b, you need to use the period of the tide cycle. The period is the time it takes for the tide to complete one cycle, which includes one high tide and one low tide.
In this case, you have the information of the high tides and low tides throughout the day. The time difference between consecutive high tides or low tides can give you the length of one cycle.
The time difference between the first high tide at 4:03 am and the next high tide at 4:25 pm is 12 hours and 22 minutes (or 742 minutes). Since this time difference corresponds to half a cycle, you need to double it to get the time for a full cycle.
So, the period (P) of the tide cycle is 2 * 742 minutes = 1484 minutes.
Now, to find the value of b, you need to use the equation: period = 2π / b.
Substituting the known value of the period, you can solve for b:
1484 = 2π / b
Multiply both sides by b:
1484b = 2π
Divide both sides by 1484:
b = 2π / 1484
Thus, the value of b is approximately 0.00425086.
Therefore, the equation for the tide cycle is y = acos(0.00425086t), where t represents the time in minutes.
Remember that the time in minutes should be measured from the starting time of the first high tide or low tide used to calculate the period.