Question

2. At the end of a dock, the high tide of 14 m is recorded at 9:00 am.
The low tide of 6 m is recorded at 3:00 pm. A sinusoidal function can model the water depth versus time.

b) Construct a model for the water depth using a sine function, where time is measured in hours past high tide.

what is the phase shift?

alright so i got amplitude = 4
The period = 30
and the vertical displacement = 10

Answer 1

This is much better. The equation for a general sine curve is \[f(x)=A\sin(Bx-C)+D\] where A is the amplitude, B is the period, C is the phase shift, and D is the vertical displacement. Your amplitude and vertical displacement are correct, but I don't know how you got 30 for the period. The period is the amount of time it takes to go from one high tide to the next, which is (in this case) 12 hours. So far, you have that \[f(x)=4\sin(12x-C)+10\] Note however, that the question states that time is measured in hours past high tide. This means that you can set your initial high tide to be at x=0, which has the high tide height of 14. Plugging these values in and simplifying a bit, you get that \[\sin(-C)=1\] This is satisfied when \[C=\frac{-\pi}{2}\] Thus, this is your phase shift.

Answer 2

Sorry, I made a slight mistake. B is equal to 2pi divided by the time it takes for one period, so B should instead be \[B=\frac{2\pi}{12}=\frac{\pi}{6}\] This does not change the phase shift however.

Answer 3

If the 30 you mentioned was because pi/6 radians is also 30 degrees, then your answer is also right. HOWEVER, you never use degrees in the equations of curves, for various reasons.

Answer 4

so what is the phase shift because the answer states that it is 3 units to the left

Answer 5

Okay, the alternate way to write sine curves is instead using the notation \[f(x)=A\sin(B(x-K))+D\] This is just factoring the above equation a bit. Thus, the equation can also be written as \[f(x)=4\sin(\frac{\pi}{6}x-\frac{\pi}{2})+10=4\sin(\frac{\pi}{6}(x-3))+10\] This is a phase shift of 3 to the left, since that is what K represents.

Answer 6

Sorry, one more little mistake, it should be x+3 in the final equation, since C is negative pi/2, rather than positive. This is a phase shift of 3 to the left.

Answer 7

The reason it is x+3 is because, since C is negative, the original function was supposed to be \[f(x)=4\sin(\frac{\pi}{6}x+\frac{\pi}{2})+10\] etc. etc.

Answer 8

i got it thanks :)
btw do I have to manipulate the value of pi for these kinds of questions as a grade 11 student. Because we havent't learned using it and I see how you used it a couple of times in your answer
'pi/2 or pi/6"

Answer 9

Well, the only reason I kept using it is that once you hit calculus, you never use degrees again (for various reasons). One of those reasons actually being that it is technically incorrect to have degrees in a function. So if you had used 30 instead of pi/6 in the above equation (for example), then it could potentially lead to incorrect answers if you use calculus on it. If you haven't learned it yet, I would just say to remember that pi is 180 degrees. If you aren't doing calculus, you can probably do everything in degrees and if the answer is a function, just convert it to radians (aka, multiples of pi) at the end.

Answer 10

alright thanks
I have calculus next year
so ill use it as a heads up