During a 12-hour period, the tides in one area of the Bay of Fundy cause the water level to rise to 6 m above average sea level and to fall 6 m below average sea level. The depth of the water at low tide is 2 m as measured against a pier.
a.) Suppose the water is at average sea level (rest position) at 0:00 hours (midnight) and the tide is coming in. Draw a graph that shows the height of the tide over a 24-hour period. Explain how you obtained the graph.
b.) b.) Write the equation of the Sine wave that represents the change in the height of the tide.
c.)If the water is at average sea level at 03:00 (3 am) instead of at midnight and the tide is coming in, write the equation of the Sine wave that represents this situation.
Appreciate any help I can get...
a.) To draw a graph that shows the height of the tide over a 24-hour period, you can use the following steps:
Step 1: Identify the intervals of high tide and low tide:
Since the tides in the Bay of Fundy cause the water level to rise 6 meters above average sea level and fall 6 meters below average sea level during a 12-hour period, we can interpret this as a full cycle of tides. Therefore, we have 6 hours of rising tide (high tide) and 6 hours of falling tide (low tide).
Step 2: Determine the maximum and minimum points of the tide:
At high tide, the water level rises to 6 meters above average sea level, while at low tide, the water level falls to 6 meters below average sea level. Considering that the depth of the water at low tide is 2 meters as measured against a pier, we can conclude that the average sea level is at 2 meters below the pier.
Step 3: Plotting the graph:
Now, let's plot the graph. We will use the x-axis to represent time and the y-axis to represent the height of the tide.
- At midnight (0:00 hours), the water is at average sea level, which is 2 meters below the pier. So, the y-coordinate is -2 meters.
- During the first 3 hours (from 0:00 to 3:00), the tide is coming in. As the graph starts to rise, the water level increases until it reaches the high tide point at 6 meters above average sea level.
- From 3:00 to 9:00, the tide is going down, causing the graph to descend. At 6:00, the water level reaches the average sea level (2 meters below the pier).
- Between 9:00 and 15:00, the tide is going out, leading to a further decrease in the water level. At 12:00 (noon), the water level reaches its lowest point, which is 6 meters below average sea level.
- From 15:00 to 21:00, the tide is coming in again. As the graph goes up, the water level gradually rises until it reaches the average sea level (2 meters below the pier) at 18:00 (6:00 pm).
- After 21:00, the tide is going down, and the water level drops below the average sea level until it reaches the low tide point at midnight.
Using these steps, you can plot the graph representing the height of the tide over a 24-hour period in the Bay of Fundy.
b.) The equation of a sine wave that represents the change in the height of the tide is generally given as:
y = A * sin(B * (x - C)) + D
Where:
- A represents the amplitude of the wave, which is half the difference between the highest and lowest points of the tide.
- B determines the period of the wave. It's given by 2π divided by the length of the cycle.
- C represents the horizontal shift of the wave, which indicates the starting point of the wave.
- D represents a vertical shift, indicating the average sea level.
Using the information given in the problem, we can determine the values of A, B, C, and D:
- A = 6 meters (half the difference between the highest and lowest points of the tide).
- B = 2π/12 (since the full cycle of the tide occurs over 12 hours).
- C = 0 (since the starting point is at midnight, which is the horizontal reference point).
- D = -2 meters (average sea level, given that the depth at low tide is 2 meters below the pier).
The equation of the sine wave representing the height of the tide is:
y = 6 * sin((2π/12) * x) - 2
c.) If the water is at average sea level at 03:00 (3 am) instead of at midnight and the tide is coming in, we need to account for the horizontal shift in the equation. In this case, the starting point of the tide is shifted by 3 hours, or 3/24 of a full cycle.
Using the same values for A, B, and D as in part b:
- A = 6 meters
- B = 2π/12
- D = -2 meters
To determine the value of C representing the horizontal shift, we multiply the period B by the fraction of the cycle (3/24) that is shifted. Therefore, C = (2π/12) * (3/24) = π/16.
The equation of the sine wave representing the height of the tide when the water is at average sea level at 03:00 is:
y = 6 * sin((2π/12) * (x - π/16)) - 2