A tidal wave reaches low tide at 3:00AM and high tide at 3:00PM.
The tide can rise and fall 10m from the sea level.
If at low tide the water level is 2m above the seabed (bottom of the sea)
a) determine an equation to model the sea level above the sea bed S and time t in hours after midnight
b). Sketch the graph for the tidal wave over a 48 hour window
Really need help with this guys :(
low tide at 3:00AM and high tide at 3:00PM
so, the period is 24 hours. Since sin(kt) has period 2π/k, we start out with
y = sin(π/12 t)
rise-fall = 10, so
y = 10sin(π/12 t)
low tide is at +2, so
y = 12+10sin(π/12 t)
We are starting out near the minimum value for t=0. cos(t) has a maximum at t=0, but we want a minimum at t=3. So, let's make that
y = 12-10cos(π/12 (t-3))
And the graph is at
http://www.wolframalpha.com/input/?i=12-10cos(%CF%80%2F12+(t-3))+for+t%3D0..24
By the way, you do know that the tide has a period of roughly 12 hours, not 24, right?
To model the sea level above the seabed (S) as a function of time (t), we can use a sine function since tides are typically periodic.
a) The general equation for a sine function is:
S = A*sin(B(t - C)) + D
Where:
A represents the amplitude (half the difference between the maximum and minimum values of the function),
B represents the frequency (how many cycles the function completes in the given time period),
C represents the phase shift (how the wave is shifted horizontally along the time axis), and
D represents the vertical shift (how the wave is shifted vertically from the equilibrium position).
In our case:
A = 10m (since the tide can rise and fall by 10m),
B = (2π/12) (since the tide completes one full cycle between 3:00AM and 3:00PM, which is 12 hours),
C = 9 (since we want the low tide to occur at 3:00AM),
D = 2m (since at low tide the water level is 2m above the seabed).
Therefore, the equation to model the sea level above the seabed S at time t in hours after midnight is:
S = 10*sin((2π/12)(t - 9)) + 2
b) To sketch the graph for the tidal wave over a 48-hour window, we can plot the function S = 10*sin((2π/12)(t - 9)) + 2, where t ranges from 0 to 48.
Here is a sketch of the graph:
^
12m + |
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10m +--------------+
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8m + |
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6m + |
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4m + |
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2m +--+-----------+-------------------->
0h 24h 48h