SOLUTION: 12. The nearby beach has an average depth of the water at low tide of 1 m at midnight. The average depth of the water at high tide is 8 m at 6am. One complete cycle takes 12 h. (a

Question 1142011: 12. The nearby beach has an average depth of the water at low tide of 1 m at midnight. The average depth of the water at high tide is 8 m at 6am. One complete cycle takes 12 h.
(a) Determine the equation that models the tides using cosine as the base function.
(b) Many people dive from this pier during the day. If the water must be at least 3 m deep to dive safely, between what daylight hours should people dive?

Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your equation will be:

y = 3.5 * cos(30 * (x + 6)) + 4.5

it looks like this.

the general form of the cosine wave is:

y = a * cos(b * (x - c)) + d

a is the amplitude
b is the frequency
c is the horizontal shift
d is the vertical shift

your amplitude is 3.5.

that's because your cosine wave will go from 8 to 1, which is a difference of 7.

divide that 2 and you get your amplitude of 3.5

that would be "a" in the general equation.

since your low tide is 1 and your high tide is 8, then your cosine wave has to go from 8 to 1.

that would put the center line at 4.5 because there are 3.5 units between 8 and 4.5 and there are 3.5 units between 4.5 and 1.

since your center line is at y = 4.5, then your vertical shift is 4.5.

that would be "d" in the general equation.

your period if the interval for one full wave.

since the wave repeats every 12 hours, then your period is equal to 12.

your frequency is equal to 360 degrees / 12 = 30.

that would be "b" in the general equation.

so far your cosine wave is 3.5 * cos(30 * x) + 4.5

your cosine wave normally starts at its high point.

that would be y = 8 in the equation as shown.

you want it to be at 1.

you can do one of two things.

if you make the amplitude negative, then the cosine wave will start at 1 rather than 8.

you can also do a horizontal shift.

since the cosine is 1 when x = 6, you need to shift it to the left 6 units so it will be y = 1 when x = 0.

that would be c in the general form of the equation.

that gets you the final equation of y = 3.5 * cos(30 * (x + 6)) + 4.5

that's what's on the graph above.

if the water needs to be at least 3 feet deep, then draw a line at y = 3 and check the times when the graph is at or above that line.

those times would be between x = 2.154 and 9.846, and between x = 14.154 and 21.846.

those hours correspond to military time which goes from 0 to 24.

x = 2.154 corresponds to somewhere between 2:14 and 2:15 am.
x = 9.846 corresponds to somewhere between 9:50 and 9:51 am.
x = 14.154 corresponds to somewhere between 2:14 and 2:15 pm.
x = 21.846 corresponds to somewhere between 9:50 and 9:51 pm.

assuming daylight is from 6:00 am to 8:00 pm, then you would get the following.

water is at least 3 feet deep between the hours of 6:00 am and 9:51 am.
water is at least 3 feet deep between the hours of 2.14 pm and 8:00 pm.

since it was not specified when the daylight hours start and stop, i assumed between 6:00 am and 8:00 pm.

you can adjust based on what you know.

Answer by ikleyn(52805) (Show Source): You can put this solution on YOUR website!
.

The formula for the 12-hours cycle is y = , where x is the measure of the time in hours, starting from x= 0 at midnight and till x= 12 hours (the noon), which is about one full tide cycle. Check. At x = 0 hours (midnight) y = = -3.5*1 + 4.5 = - 3.5 + 4.5 = 1 meter (lowest tide level); at x= 3 hours y = = = = = 4.5 meters; at x = 6 hours y = = = = = 8 meters (highest tide level); at x = 9 hours y = = = = 4.5 meters; at x = 12 hours (at noon) y = = = -3.5*1+ 4.5) = -3.5 + 4.5 = 1 meter (lowest tide level, again). The period (one full cycle) is equal to 12 hours: = , as it should be.

You can use the same formula for the two consecutive full tide cycles (of the total 24 hours duration), too.

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To get familiar with the subject, read this Internet article

https://oceanservice.noaa.gov/education/tutorial_tides/tides05_lunarday.html

Because the Earth rotates through two tidal “bulges” every lunar day, coastal areas experience two high and two low tides every 24 hours and 50 minutes. High tides occur 12 hours and 25 minutes apart. It takes six hours and 12.5 minutes for the water at the shore to go from high to low, or from low to high.

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