a We are told that the low tide at a port is 3.5 feet and occurs at midnight, which is represented by t=0. After 6 hours the port is at high tide, which is 16.5 feet.
We want to write a sinusoidal model for the tide depth d at the port, as a function of the time t. To do so, we will consider the general forms of transformed
Sine Function:& y= asin b(t- h)+ k
Cosine Function:& y= acos b(t- h)+ k
In these functions | a| is the , 2Ď€ b is the ( b>0), h is the horizontal , and k is the vertical shift. To write a function for a sinusoid, we will follow four steps.
- Use the to identify any vertical shift k.
- Determine whether the tide depth should be modeled by a sine or a cosine function, and identify any horizontal shift h.
- Find the period and the value of b.
- Find the amplitude and the value of a.
Let's do it!
Step 1
The vertical shift k is the of the maximum and minimum values.
k=maximum value+ minimum value/2
We know that the maximum value is 16.5 and the minimum value is 3.5.
k=16.5+ 3.5/2
Let's evaluate this formula to find the value of k.
k = 16.5+3.5/2
k = 20/2
k= 10
Step 2
Since the minimum point occurs at t=0, the function passes through y-axis at its minimum point. Then, the tide depth can be modeled as a cosine function across the x-axis and with no horizontal shift. This means that h= 0.
Step 3
We know that six hours after midnight the port is at high tide. This represents the half of one cycle. To find the period we need to multiply six hours by two, because period corresponds to one whole cycle.
Period: 2(6) = 12
We also know that the period of sine and cosine functions is 2Ď€ b. With this information, we can write an equation in terms of b.
12=2Ď€/b
Let's solve this equation!
We found that b= π6.
Step 4
Next, let's find the amplitude | a|. This is half the difference between the maximum and minimum values.
| a|=maximum value- minimum value/2
In our case, the maximum value is 16.5 and minimum value is 3.5.
| a|=16.5- 3.5/2
We will solve this equation to find | a|. Let's do it!
|a| = 16.5-3.5/2
|a| = 13/2
|a| = 6.5
Equation of the Model
Because of the location of the minimum point, we concluded that the tide depth should be modeled by a cosine function. We also found that a= - 6.5, b= π6, h= 0, and k= 10. Finally, we can write the equation of the model.
d = - 6.5 cos π/6( t - 0)+ 10
⇕
d=- 6.5 cos π/6 t + 10
