The water level in a lake rises and falls throughout the day. The average depth of the water at low tide is 4 metres, while the average depth at high tide is 8 metres. It takes 6 hours for the water level to rise from low tide to high tide.
a) Determine a cosine equation to model the water depth, assuming that at t = 0
hours the water is at low tide
b) What is the depth of the water 2 hours after low tide?
Amplitude is (max-min)/2 = (8-4)/2 = 2
middle line is (max+min)/2 = 6
the period is roughly 12 hours, so 2π/k = 12
So far, we have y = 6+2cos(π/6x)
Since cos(x) is max at t=0, and we want it to be a min, make that
y = 6-2cos(π/6 x)
Now plug in x=2
To determine a cosine equation to model the water depth, we can use the equation for a cosine function:
y = A*cos(B(x-C)) + D
Where:
- A represents the amplitude, which is half the difference between the maximum and minimum values of the function.
- B represents the frequency, which is 2π divided by the period of the function.
- C represents the phase shift, which is the horizontal shift of the function.
- D represents the vertical shift, which is the shift up or down of the function.
In this case, the amplitude is half the difference between the maximum depth (8m) and the minimum depth (4m), which is (8 - 4)/2 = 2m.
The frequency can be determined by the period, which is the time it takes for the water level to go from low tide to high tide, in this case, 6 hours. Therefore, the frequency is 2π/6.
As the water is at low tide at t = 0, we don't need a phase shift, so C = 0.
The vertical shift, D, is the average depth at low tide, which is 4m.
Putting all this information together, we can write the equation for the water depth as:
y = 2*cos((2π/6)x) + 4
Now, to find the depth of the water 2 hours after low tide, we can substitute x = 2 into the equation:
y = 2*cos((2π/6)*2) + 4
Simplifying the equation:
y = 2*cos((4π/6)) + 4
Evaluating the cosine function:
y = 2*cos((2π/3)) + 4
Using a calculator, we can find that cos((2π/3)) is approximately -0.5:
y = 2*(-0.5) + 4
Simplifying the equation:
y = -1 + 4
y = 3
Therefore, the depth of the water 2 hours after low tide is 3 meters.