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, we need to understand the pattern of the tides in the Bay of Fundy. We know that the tides cause the water level to rise to 6 m above average sea level and fall 6 m below average sea level.
At 0:00 hours (midnight), the water is at average sea level (rest position). From there, we can observe that every 6 hours, the tide reaches its peak, either 6 m above average sea level or 6 m below average sea level. So, we can mark the first peak at 6:00 hours (18:00 hours on the previous day) at 6 m above average sea level.
Then, at 12:00 hours (noon), we will have the second peak, at 6 m below average sea level.
To complete the pattern, we need to add the low tide points. We know that the depth of the water at low tide is 2 m as measured against a pier. Therefore, we can mark the low tide points at 3:00 hours (15:00 hours on the previous day) and 9:00 hours (21:00 hours on the previous day) at 2 m below average sea level.
This creates a repeating pattern where the tide goes from average sea level to 6 m above, then back to average sea level, then 6 m below, and then back to average sea level again. The graph will show this pattern repeating every 12 hours.
b.) The equation of a sine wave that represents the change in the height of the tide can be written as follows:
h(t) = A * sin(B * t + C) + D
Where:
- A represents the amplitude, which is the maximum deviation from the average sea level. In this case, A = 6 since the tides can rise up to 6 m above and fall 6 m below average sea level.
- B represents the frequency, which is the number of complete cycles per unit of time. In this case, the frequency is 1 cycle per 12 hours or B = (2 * π) / 12.
- C represents the phase shift, which determines the starting point of the wave. In this case, since the tide starts at average sea level at 0:00 hours, C = 0.
- D represents the vertical shift, which adjusts the average sea level. Since the tide rises and falls 6 m above and below average sea level, the vertical shift is 0, or D = 0.
Therefore, the equation of the sine wave representing the change in the height of the tide is:
h(t) = 6 * sin((2 * π / 12) * t)
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 adjust the starting point or phase shift of the wave.
Since the tide is starting at 03:00 hours, we need to shift the wave 3/12 or 1/4 of a cycle to the right. This means we need to subtract a phase shift of (π / 2) from the original equation. Therefore, the equation of the sine wave representing this situation becomes:
h(t) = 6 * sin((2 * π / 12) * t - (π / 2))