Here's a neat problem that I made for my Pre-Calculus 12 class in Shanghai China which relates a sinusodal function graph to tides and a beautiful ship on which I've sailed and taught students! The Sail and Life Training Society's Pacific Swift.
Here's a link to an online graphing calculator called Desmos which I use in almost all my lessons and a few screenshots from a TI-83 calculator emulator called Wabbit to help you understand how to do the question!
Sailboats like the Pacific Swift have deep keels to help them track a straight line through the water despite being pushed sideways by the wind. The heavy keel also helps them to stay upright instead of being tipped over to one side (called heeling over). The deep keel, however, can be a bit of a liability when navigating close to shore, so sailors need to know understand the fluctuating tidal cycles to determine the exact depth of the water and avoid running aground.
The tides in the inner harbour in Victoria BC follow a sinusoidal variation.
The low tide at 12:00am is 6ft deep.
The next high tide is at 12:00pm is 24ft deep.
The next low tide is 6ft at 12:00am.
a) Determine an equation h(t) for the function of the height of the water h in terms of time t hours.
b) The Pacific Swift, a classic Canadian sailboat, requires at least 10 feet of water to enter the harbour. Determine the range of times in the day that the Pacific Swift can enter the harbour.
Here's how to do it!
First let's take a look at the 4 components of a Sin function:
A sinusodal function has the equation f(x) = aSin b(x-c) + d
a - is equal to the amplitude of the function
b - relates to the period of the function with the equation Period = (2π)/b
c - is the phase shift of the function (also called the horizontal translation)
d - is the vertical translation of the function
Let's start with a sketch to see what this graph looks like!
For the x-axis we'll use the values for time of day from the question (12:00am to 12:00am)
For the y-axis we'll use water depth ( 0ft - 24ft)
Here's a link to an online graphing calculator called Desmos which I use in almost all my lessons and a few screenshots from a TI-83 calculator emulator called Wabbit to help you understand how to do the question!
Sailboats like the Pacific Swift have deep keels to help them track a straight line through the water despite being pushed sideways by the wind. The heavy keel also helps them to stay upright instead of being tipped over to one side (called heeling over). The deep keel, however, can be a bit of a liability when navigating close to shore, so sailors need to know understand the fluctuating tidal cycles to determine the exact depth of the water and avoid running aground.
The tides in the inner harbour in Victoria BC follow a sinusoidal variation.
The low tide at 12:00am is 6ft deep.
The next high tide is at 12:00pm is 24ft deep.
The next low tide is 6ft at 12:00am.
a) Determine an equation h(t) for the function of the height of the water h in terms of time t hours.
b) The Pacific Swift, a classic Canadian sailboat, requires at least 10 feet of water to enter the harbour. Determine the range of times in the day that the Pacific Swift can enter the harbour.
Here's how to do it!
First let's take a look at the 4 components of a Sin function:
A sinusodal function has the equation f(x) = aSin b(x-c) + d
a - is equal to the amplitude of the function
b - relates to the period of the function with the equation Period = (2π)/b
c - is the phase shift of the function (also called the horizontal translation)
d - is the vertical translation of the function
Let's start with a sketch to see what this graph looks like!
For the x-axis we'll use the values for time of day from the question (12:00am to 12:00am)
For the y-axis we'll use water depth ( 0ft - 24ft)
 |  |
Now let's connect the dots with a nice smooth sinusodal line!
Now that we have a visual reference to help us, let's find the 4 variables that will help us solve the equation!
The first variable to search for is "a" the amplitude, we do this by taking the maximum value (high tide at 24ft) subtracting the minimum value (low tide at 6ft) and dividing by 2.
The first variable to search for is "a" the amplitude, we do this by taking the maximum value (high tide at 24ft) subtracting the minimum value (low tide at 6ft) and dividing by 2.
The second variable to find is "d" the vertical translation; this is the centre line that the tide will fluctuate above and below. Once we know the amplitude it's as easy as taking the minimum value and adding the amplitude. (You could also take the maximum value and subtract "a")
The third "thing" to find is the period. This will help us to determine the value of the variable "b"
The period is defined as the the time it takes for the graph to go through one complete cycle.
In our question, the low tide happens at 12:00am and the next high tide happens at 12:00pm. That's half our period.
Our whole period will be from low tide to the next low tide which happens at 12:00am, 24 hours later.
So our period is 24 hours. Now let's see how we use our Period to find "b"
The period is defined as the the time it takes for the graph to go through one complete cycle.
In our question, the low tide happens at 12:00am and the next high tide happens at 12:00pm. That's half our period.
Our whole period will be from low tide to the next low tide which happens at 12:00am, 24 hours later.
So our period is 24 hours. Now let's see how we use our Period to find "b"
Okay so now we've found "a", "b" and "d" but what about that pesky "c" value?
Well "c" (the phase shift) depends on whether we are using a Sine function or a Cosine function. Let's take a look a these functions to see their shapes!
Head over to Desmos online graphing tool and check out the shapes off all the trig functions!
So we can see that:
f(x) = Sin x starts at (0,0) and moves up towards a maximum
f(x) = Cos x starts at the maximum (0,1) and moves down towards the minimum
That doesn't seem to match our graph at all, but if we flip f(x) = Cos x upside down by reflecting it in the x-axis we can see that:
f(x) = -Cos x starts at the minimum (0,-1) and moves down towards the maximum
So, just to recap
a = 9
b = Pi/12
c = 0 if we use a negative cosine graph
d = 15
And when we put it all together:
Well "c" (the phase shift) depends on whether we are using a Sine function or a Cosine function. Let's take a look a these functions to see their shapes!
Head over to Desmos online graphing tool and check out the shapes off all the trig functions!
So we can see that:
f(x) = Sin x starts at (0,0) and moves up towards a maximum
f(x) = Cos x starts at the maximum (0,1) and moves down towards the minimum
That doesn't seem to match our graph at all, but if we flip f(x) = Cos x upside down by reflecting it in the x-axis we can see that:
f(x) = -Cos x starts at the minimum (0,-1) and moves down towards the maximum
So, just to recap
a = 9
b = Pi/12
c = 0 if we use a negative cosine graph
d = 15
And when we put it all together:
Check out this link to Desmos to see how it all fits together!
Andrew's Lesson Time and Tides
Andrew's Lesson Time and Tides
Now comes the fun part! We've got our function that tells how the time of day relates to the depth of the water, so now we can determine when the water is deep enough for the Pacific Swift to safely navigate the harbour.
If the minimum water depth needed for the Pacific Swift to enter or leave the harbour is 10ft then we need to add a second function to our question y = 10. We'll use technology to help us find the exact time!
If the minimum water depth needed for the Pacific Swift to enter or leave the harbour is 10ft then we need to add a second function to our question y = 10. We'll use technology to help us find the exact time!
 Make sure you use brackets carefully! |  Check that your window range is within the limits of your domain and range. |  Oooohhhh pretty. The part of the function that is above y = 10 is when the water is deep enough. |
Now we use "2nd function - Calc - Intersect" to find the two places where the graphs meet.
The first intersection is when x = 3.75
The second intersection is when x = 20.25
That's fine, but if you told the Captain that it was safe to enter the harbour at 3.75hrs he might look at you as if you had cabin fever.
Let's convert 3.75 hours into hours and minutes.
3 hours plus (0.75 hours x 60 minutes/hour ) = 3:45am
It's safe to enter the harbour at 3:45am
Now let's find 20.25 hrs.
20hrs is 8pm and (0.25 hours x 60 minutes/hour) = 15 minutes
It's safe to leave the harbour before 8:15pm.
So a quick stop into the inner harbour for lunch at Red Fish Blue Fish and we'll be back on the water again!
Stop by again soon to see how this lesson relates to DJ'ing!
Drew
The first intersection is when x = 3.75
The second intersection is when x = 20.25
That's fine, but if you told the Captain that it was safe to enter the harbour at 3.75hrs he might look at you as if you had cabin fever.
Let's convert 3.75 hours into hours and minutes.
3 hours plus (0.75 hours x 60 minutes/hour ) = 3:45am
It's safe to enter the harbour at 3:45am
Now let's find 20.25 hrs.
20hrs is 8pm and (0.25 hours x 60 minutes/hour) = 15 minutes
It's safe to leave the harbour before 8:15pm.
So a quick stop into the inner harbour for lunch at Red Fish Blue Fish and we'll be back on the water again!
Stop by again soon to see how this lesson relates to DJ'ing!
Drew
