Question

Tides are cyclical phenomena caused by the gravitational pull of the sun and the moon. On a particular retaining wall, the ocean generally reaches the 3 m mark at high tide. At low tide, the water reaches the 1 m mark. Assume that high tide occurs at 12:00 p.m. and at 12:00 a.m., and that low tide occurs at 6:00 p.m. and 6:00 a.m. What is the height of the water at 10:30 a.m.?

Answer 1

Assume that the cyclic phenomenon can be described by a sinusoidal curve.
If we set t = 0 at 12:00 pm, we can use the following formula:
\[H(t) = a + b \cos (\frac{ 2 \pi }{ T } t)\]
A cos function is easier than a sine, because the curve starts at t = 0 in the max. value (high tide) and the cos function also starts at max value.

Now we have to determine the parameters a, b and T.
a = value between high and low, so a = ?
b = amplitude = ?
T = period of phenomenon, so T = 12.

If you realize what part of the period 10:30 means, you could even get an exact value of H.

But first, try to find a and b!

Answer 2

could you please show me how to find a and b and then show me how to get the exact answer, basically i mean could you do this entire question for me so that i can learn. thank you.

Answer 3

High tide is 3, low tide is 1, so in the middle is 2, that's a.
b is the amplitude, this is the distance from the middle line (2) to the min or max. Here we have 2+1 (max) and 2-1 (min). So b = 1.
T is the period. We have a cos-function. If you look at the graph of a cos function, you will see that it begins in het max value, goes all the way down to the min value and at the end it is max again. That's one period. In the story of the tides it says: max at 12 pm and again at 12 am, so 12 hours. T = 12.
All in all:\[H(t)=2+\cos \left( \frac{ 2 \pi }{ 12 }t \right)=2+\cos \left( \frac{ 1 }{ 6 }\pi t \right)\]
Hope this helps!

Answer 4

but you still did not answer what it is actually asking, which is What is the height of the water at 10:30 a.m.? could you please do the entire answer. thank you so much.

Answer 5

I forgot that one :(

The trick is, you don't even put T = 10.5 (!) in the formula, because 10:30 hours is equal to 7/8 of the period. See attached picture of a circle (should be unit circle, but never mind).
I hope you are familiar with some well-known values of sin and cos, in this case it is cos(1.75pi). (7/8 of 2pi is 1.75pi). It is just the same as cos(0.25pi). It is 0.5sqrt(2).
So the exact height of the water at 10:30 is \[H(10.5)=2+\frac{ 1 }{ 2 }\sqrt{2} (\approx 2,707) \]

Answer 6

could you write all the steps, and the whole response again, clearly so i can see it? sorry its kind of all over the place so its hard for me to understand.