giving sin graph an equation word problem

I am in precalculus and practicing problems from my textbook. I came across this question that I haven't been able to solve.

Here is the question:

High tide at the local beach occurs at 4:00am, when the depth of the water is 11 feet. 
Low tide occurs at 10:00am, when the depth of the water is 4 feet. Let t=0 represent midnight. 
Write the equation that represents the tide as a function of time.

The answer I came up with was

y = 3.5sin(π/6t)+7.5

But this is not right. My thinking is that the amplitude is 3.5, the midline is 7.5, and the period is π/6 because a full period is 12 hours.

Any idea on what I am doing wrong?

Solution 1:

Let $t$ represent hours since midnight and $y$ represent the wave's height in feet, assumed to be sinusoidal in time. You got the amplitude and vertical shift and period of the sine wave right. To get the phase shift, note that the wave is at its midline at (4+10)/2=7 am. Note that the height is decreasing as we cross this time so we throw in a negative sign in front of the amplitude. So we have

$$y=-3.5\sin\left(\frac{\pi}{6}(t-7)\right)+7.5.$$

You will see this works if you plug in the given information, and drawing a pic can also help. You may also find this helfpul.