At the ocean, it is known that the tide follows a trigonometric path. At high tide,
the water comes in to a point 1 meter from where I placed a flag. At low tide, the
water comes in to a point 11 meters from the same flag. The time it takes from to
get from high tide to low tide is 5 hours. It is now midnight and it is high tide.
(Note: low tide=max and high tide=min in this case)
Plot the motion for two complete cycles and state a possible equation for this motion.
I have absolutely no idea where to even begin with this problem, other than the equation involves either sin(x) or cos(x). Any tips on where to start or help working out how to do this word problem would be much appreciated.
assume it is
distance=Asin(wt+theta) + k
low tide:
11=Asin(w5+theta)+k
high tide
1=Asin(0+theta)=k
but you also know that that at t=5, half a cycle has occured, to A= (11-1)/2
high tide:
1=5sin(theta)+k
low tide
11=5sin(PI + theta) + k
but sin(PI+theta)=-sinTheta so
11=-5sinTheta +k
add the equations for high tide and low tide
12=2k k=6
subtract the equations
10=10sinTheta
theta= PI/2
so you have theta, A, k
To solve this problem, we need to understand the relationship between the position of the tide and time. Let's break down the information given:
1. At high tide, the water comes in to a point 1 meter from the flag.
2. At low tide, the water comes in to a point 11 meters from the flag.
3. The time it takes for the tide to go from high tide to low tide is 5 hours.
4. It is currently midnight, and it is high tide.
To visualize the motion, we can plot the position of the tide as it varies with time.
Let's assume that the horizontal distance from the flag to the water is represented by the variable x, with positive values indicating the water is getting closer to the flag, and negative values indicating that it's moving away.
1. At high tide, the water is 1 meter from the flag, so at t=0 hours, x=1.
2. At low tide, the water is 11 meters from the flag, so at t=5 hours, x=-11.
3. The motion during one complete cycle would go from high tide to low tide and then back to high tide. Therefore, we need to consider two complete cycles, which would take 10 hours (2 cycles of 5 hours each).
4. Since it is currently high tide at midnight, our time interval will be from 0 to 10 hours.
To find a possible equation for this motion, let's start by considering a sine function.
A general equation for a sine function is:
y = A*sin(Bx + C) + D,
where:
- A represents the amplitude (the difference between the highest and lowest points of the wave),
- B determines the period (the time it takes for the wave to complete one full cycle),
- C represents a phase shift (it determines where the wave starts),
- D is a vertical shift (it moves the whole wave up or down).
Since we are dealing with the tide going from high tide (maximum) to low tide (minimum), the amplitude is (11-1)/2 = 5.
To determine the period, we know that it takes 5 hours to go from high tide to low tide. Therefore, the time for one complete cycle is 10 hours. This gives us a B value of 2π/10 = π/5.
Since we want the high tide to occur at t=0, there is no phase shift, so C=0.
Finally, since it is currently high tide at midnight, we need a vertical shift of 1 meter, so D=1.
Putting it all together, the equation for the motion of the tide could be:
x = 5*sin((π/5)t) + 1.
This equation represents the position of the tide as it varies with time. To plot the graph of this motion, you can use software like Excel or online graphing tools, entering the equation and specifying the time interval from 0 to 10 hours.
Remember, this equation is just one possible solution based on the given information. There could be other equations that also describe the same motion.