A cruise ship is docking in Triga Bay.

Question

A cruise ship is docking in Triga Bay. The depth of the water changes due to the tides. At 6 am it is low tide and the water is 2 m deep. At noon it is high tide, and the water is 8 m deep. 

Draw a sinusoidal graph to represent the depth of the water as a function of the hours since midnight for a full day.

Find the equation of this function. Use to find the k value.

Use your equation to determine the depth of the water at 4 pm.

Answer 1

Ok, so we know that the minimum value is 2m, and the maximum is 8m. This means that the amplitude of the sine wave is (8-2)/2=3m, and the wave oscillates around 2+3=5m of height.

The wave goes from minimum to maximum in 6 hours (6am to noon), so the full cycle will take 12 hours and we will have two cycles in a full day. We are told to use midnight as our zero-time, and it will be the same as noon and the wave will be at the maximum and starting to decrease. This sounds like a cosine function, or sine offset by a quarter cycle. Since our input is in terms of hours, we'll need to multiply it by 360/12=30 since a normal cycle would be 360 degrees and we are completing it in 12 hours.

I'll use h to represent hours since midnight.

Putting it all together, our equation will be depth = 3m * cos(30*h) + 5m or, equivalently, 3m*sin(30*(h + 3)) + 5m = 3m*sin(30h+90) +5m. Our k-value is 30 (from 360/12) if we're using degrees, or pi/6 from (2*pi/12) if we're using radians.

At 4pm, the water will be 3m*cos(30*16)+5m = 3m*(-0.5)+5m = 5m - 1.5m = 3.5m