Question

At a particular location on the Atlantic coast a pier extends over the water. The height of the water on one of the supports is 5.4 feet, at low tide (2am) and 11.8 feet at high tide, 6 hours later. (Let t = 0 at midnight)

a) Write an equation describing the depth of the water at this location t hours after midnight.

Answer 1

h(t)=-3.2cos[(2pi/12.4)(t-2)]+8.6

Explanation:

The depth of the water can be modeled using a cosine function of the form h(t) = A*cos(B(t-C))+D, where h(t) represents the depth of the water at time t, A is the amplitude, B determines the period, C is the horizontal shift, and D is the vertical shift.

First, we can find the amplitude A by taking half the difference between the maximum and minimum values. In this case, the maximum value is 11.8 feet and the minimum value is 5.4 feet, so A = (11.8 - 5.4)/2 = 3.2.

Next, we can find the vertical shift D by taking the average of the maximum and minimum values. In this case, D = (11.8 + 5.4)/2 = 8.6.

The period of a cosine function is given by 2π/B, where B is the coefficient of t in the argument of the cosine function. In this case, we know that high tide and low tide occur every 12.4 hours apart, so the period is 12.4 hours. Therefore, we can find B by solving for it in the equation 12.4 = 2π/B, which gives us B = 2π/12.4.

Finally, we need to find the horizontal shift C. We know that at low tide (2am), the depth of water is at its minimum value (5.4 feet). Since low tide occurs 2 hours after midnight (t=0), we can find C by solving for it in the equation h(2) = 5.4. Substituting in all known values and solving for C gives us:

• h(2) = -3.2*cos[(2π/12.4)(2-C)]+8.6 = 5.4
• -3.2*cos[(2π/12.4)(2-C)] = -3.2
• cos[(2π/12.4)(2-C)] = 1
• (2π/12.4)(2-C) = 0
• C = 2

So, putting it all together, we get that an equation describing the depth of water at this location t hours after midnight is:

h(t)=-3.2*cos[(2π/12.4)(t-2)]+8.6