The water depth in a harbor is 12m at high tide and 4m at low tide. One cycle is completed every 12 hours.
a) Sketch the graph comparing the depth of the water and time for a 24-hour period, starting a low tide.
b) Determine an equation of a cosine function that represents the water depth, in meters, as a function of the time.
Cannot sketch on these posts.
amplitude = (12-4)/2 = 4
center line is at y=(12+4)/2 = 8
period = 12, so k = π/6
minimum is at t=0, so
y = 8 - 4cos(π/6 t)
a) To sketch the graph comparing the water depth and time for a 24-hour period, starting at low tide, follow these steps:
1. On a piece of graph paper, draw the x-axis horizontally and label it "time (hours)".
2. Draw the y-axis vertically and label it "depth (meters)".
3. Mark the x-axis at regular intervals to represent the 24-hour period. You can use increments of 3 hours, or any other suitable interval.
4. Mark the y-axis to represent the possible depths of the water. Since the water depth ranges from 4m to 12m, you can mark increments of 2m on the y-axis.
5. At time 0 (the start), mark a point on the graph at the low tide depth of 4m.
6. As the time progresses and reaches 6 hours, mark a point on the graph at the high tide depth of 12m.
7. Continue marking points on the graph every 6 hours, alternating between the low tide (4m) and high tide (12m) depths.
8. Connect the marked points on the graph with a smooth curve.
The resulting graph will show a sinusoidal curve oscillating between the low tide and high tide depths over the 24-hour period.
b) To determine an equation of a cosine function that represents the water depth as a function of time, you can use the general equation for a cosine function:
y = A * cos(B * (x - C)) + D
- A represents the amplitude, which is half the difference between the maximum and minimum values. In this case, A = (12m - 4m)/2 = 4m.
- B represents the period, which is the length of one complete cycle. Since one cycle is completed every 12 hours, B = 2π/12 = π/6.
- C represents the phase shift, which is the horizontal shift of the function. Since the low tide occurs at time 0, C = 0.
- D represents the vertical shift, which moves the cosine graph up or down. Since the low tide depth is 4m, D = 4m.
Putting it all together, the equation becomes:
y = 4 * cos((π/6) * x) + 4
So, the equation that represents the water depth as a function of time is y = 4 * cos((π/6) * x) + 4.
a) To sketch the graph comparing the depth of the water and time for a 24-hour period, starting at low tide, follow these steps:
1. Label the x-axis as time in hours and the y-axis as water depth in meters.
2. Mark the time intervals at regular intervals on the x-axis, starting from 0 and going up to 24 hours.
3. Mark the water depth intervals on the y-axis, starting from the lowest point, which is 4m, and going up to the highest point, which is 12m.
4. Since one cycle is completed every 12 hours, the low tide repeats every 12 hours.
- At t = 0 hours, mark the water depth as 4m.
- At t = 12 hours, mark the water depth as 4m again.
- At t = 24 hours, mark the water depth as 4m, completing the cycle.
5. The high tide occurs halfway between the low tides, so mark the highest water depth, which is 12m, at t = 6 hours and t = 18 hours.
6. Connect the points smoothly to form a curve that represents the water depth over time. The curve should start at 4m, rise to 12m, and then descend back to 4m.
b) To determine an equation of a cosine function that represents the water depth as a function of time, follow these steps:
1. Let t represent time in hours since the start of the 24-hour period.
2. Define the amplitude of the cosine function as half the difference between the highest and lowest water depths, which is (12m - 4m) / 2 = 4m.
3. Define the period of the cosine function as the time it takes to complete one cycle, which is 12 hours.
4. Since we want the cosine function to start at the lowest tide, we need to shift it horizontally to the right by half a cycle, which is 6 hours.
5. The general equation for a cosine function is A * cos(B(t - C)) + D, where A represents the amplitude, B is the frequency (2π/period), C is the horizontal shift, and D is a vertical shift.
6. Plugging in the given values, the equation for the water depth as a function of time is: D(t) = 4 * cos((2π/12)(t - 6)) + 8
Note: The equation assumes the depths are measured from the mean sea level or a fixed reference point.