Tide Word Problem (Trigonometry)

Question

The water at a boat dock is 7 feet deep at low tide, and 11 feet deep at high tide. One day, low tide occurred at 4 am and high tide at 10 am. Find a function for the height of the tide as a function of time t .

Think about the characteristics: amplitude, period, midline, and try to sketch a figure reflecting how the tide height moves up and down.

Answer 1

Marlenny:

We will try to find an equation of the form y(t) = A*cos(ω(t-C))+B

where A is amplitude, B is vertical shift, period = (2*π)/ω, C is horizontal offset.

We know the maximum height is y=11 and the minimum height is y =7 so the amplitude is A= (1/2)*(11-7)=2

Or, the tide bounces up and down +/- 2 from y = 9. So B = 9

period = 12 (12 hours to return to the same tide level)

so ω = (2*π)/12 = π/6

The tricky part is the horizontal offset. We know that the max tide occurs at 4 am. Think about the cos(x) graph. The maximum value of 1 occurs at x = 0. Since our max occurs at t=4 it makes sense to use C = 4 as the horizontal offset.

y(t) = 2*cos((π/6)*(t-4)) + 9 gives you the tide. Try plugging in t=10, and t=16 (12 hours after 4) to make sure you get the required values. If you have any questions please ask me!

Answer 2

The amplitude is the half of the top minus the bottom: A= (11-7)/2=2.

Assuming it takes 6 hours to go up and then 6 hours to go down, a full period is 12 hours. We want the inside of our function to do "one lap" in 12 hours. One lap of a sine/cosine function is 2π, therefore we want the inside to go from 0 to 2π as time t goes from 0 to 12 hours. That is, when t=12, then Wt should equal 2π. That is, solve the following for the frequence W

W(12)=2π

W = π/6.

Now that we have the amplitude and the frequency, let's decide on whether to choose sine or cosine. Since we are starting at 4 am and this is the bottom of the tide, we should choose the function which starts low at the bottom and rises, but there isn't one. Instead, we can choose cosine, which starts at the top and falls, then multiply it by negative one. So far we have that the tide's height is

h = -2cos(pi/6 t).

However, the midline of the tide is halfway between the top and bottom, or (7+11)/2=9, so we need to shift the height up by 9 feet. Now we have

h = -2cos(pi/6 t) + 9.

By choosing cosine in this way, we don't have to worry about a phase shift.

We start the cycle at 4 am, which means that t=0 at 4 am. Then t=6 at 10 am. Check your work by calculating the height at 4 am, 10 am, and a couple of times in between. Here's a picture which will also help you confirm: https://www.desmos.com/calculator/mmz5kqh8pq