The water level (y) in a tank oscillates between 4 feet and 10 feet. It takes 20 seconds for the water level to go from 4 feet to 10 feet and seconds to go from 10 feet to 4 feet. At time(t)=0 seconds, the water level (y) is 10 feet.
1. Write the equation for the depth in the form of y=A cos B (t- C)+D, where t=time in seconds and y=depth in feet.
2. Write the equation for the depth in the form of y=A sin B (t- C)+D, where t=time in seconds and y=depth in feet.
since y is a max when t=0, we want a cosine function, since cos(0) = 1.
y = 7+3cos(pi/20 t)
because the period is 40 seconds, and the level varies between 7+3 and 7-3 feet..
C=0 because cosine is a max at t=0, so there is no offset.
Naturally, you can now write the sine function, since cos(x) = sin(pi/2 - x)
how can i put that in the form of A cos B (t- C)+D and A sin B (t- C)+D?
surely you can see that
A=3
B=pi/20
C=0
D=7
I mean, just read it off.
Now employ your skills to convert your cos(x) to sin(pi/2-x)
A and D stay the same.
To find the equations for the depth (y) in the tank, both as a cosine function and a sine function, we need to understand the properties of the oscillation.
1. Equation in the form of y = A cos B (t - C) + D:
The general equation for a cosine function is given as y = A cos(Bt + C) + D, where:
- A represents the amplitude, which is half the difference between the maximum and minimum values of y.
- B represents the frequency, which is equal to 2π divided by the period of the function.
- C represents the phase shift, which indicates any horizontal translation of the function.
- D represents the vertical shift or the average value of the function.
Given that the water level oscillates between 4 feet and 10 feet, the amplitude is half of the difference between these values, which is (10 - 4) / 2 = 3. In this case, A = 3.
The period of the function, which is the time it takes for one complete oscillation, can be calculated based on the given information. It takes 20 seconds for the water level to go from 4 feet to 10 feet, and the same time for it to go from 10 feet back to 4 feet. Thus, the total period is 20 + 20 = 40 seconds. This means that B in our equation will be 2π divided by the period, which is 2π / 40 = π / 20.
The phase shift, C, is given as the time when the water level is at its maximum value, which is at t = 0 seconds. Therefore, C = 0.
Since at t = 0 seconds, the water level is 10 feet, it implies that D = 10.
Combining all the values we obtained, the equation for the depth in the form of y = A cos(Bt - C) + D is:
y = 3 cos((π/20)t) + 10.
2. Equation in the form of y = A sin B (t - C) + D:
Similarly, the equation for a sine function can be represented as y = A sin(Bt + C) + D.
Using the same values obtained above, the equation for the depth in the form of y = A sin(Bt - C) + D is:
y = 3 sin((π/20)t) + 10.