The average depth of the water in a port on a tidal river is 4 m. At low tide, the depth of the water is 2 m. One cycle is completed approx every 12 h. a) find an equation of the depth d(t) metres, with respect to the average depth, as a function of the time, t hours, after low tide, which occurred at 15:00.
Since it asks for depth wrt average, we do not need the 4 m average but only how much it departs from average
at t = 0, d = -2 (low tide, 2 feet below average)
d = -2 cos 2 pi t/T
since T = 12 hours is given
d = -2 cos (pi t/6)
ok thanks a lot for your help :D
To find an equation for the depth d(t) in meters, with respect to the average depth, as a function of the time t hours after low tide, we can use a trigonometric function to model the cyclic nature of the tides.
First, let's determine the period of the tide cycle. We are told that one cycle is completed approximately every 12 hours. Since low tide occurred at 15:00, we can denote the start of the cycle as t = 0 when it's 15:00. Therefore, one cycle is completed when t = 12.
Next, let's determine the amplitude of the tide. The average depth of the water is 4 meters, and at low tide, it's 2 meters. The difference between the average depth and low tide is 4 - 2 = 2 meters. Therefore, the amplitude of the tide is 2 meters.
Given this information, we can use the cosine function to model the depth of the water at any given time t hours after low tide:
d(t) = A * cos(2π(t - t0) / T) + D
Where:
- A represents the amplitude of the tide (2 meters in this case)
- t0 represents the starting time of the cycle (15:00 in this case, which translates to t = 0)
- T represents the period of the tide cycle (12 hours in this case)
- D represents the average depth of the water (4 meters in this case)
Finally, substituting the values into the equation, we get:
d(t) = 2 * cos(2π(t - 0) / 12) + 4
Simplifying further:
d(t) = 2 * cos(π(t / 6)) + 4
So, the equation for the depth d(t) is 2 * cos(π(t / 6)) + 4, where t represents the number of hours after low tide occurred at 15:00.
To find an equation of the depth as a function of time, we need to consider the periodic nature of the tides. Tides follow a predictable pattern, with high tide occurring every 12 hours and low tide occurring midway between the high tides.
Let's break down the problem step by step:
1. We know that the average depth of the water is 4 meters, and the depth at low tide is 2 meters. Hence, the maximum change in depth from low tide to high tide is 4 - 2 = 2 meters.
2. Since one cycle is completed approximately every 12 hours, the time it takes for the depth to go from low tide to high tide (or vice versa) is half of that, which is 6 hours.
3. We need to find an equation that represents the depth (d) as a function of time (t) in the form d(t) = average depth + change in depth * cosine(2π/period * (t - phase shift)).
4. In this case, the average depth is 4 meters, the change in depth is 2 meters, the period is 6 hours, and the phase shift is the time elapsed since low tide, which occurred at 15:00 (3:00 PM), so it is t - 15.
Putting it all together, the equation of the depth with respect to the average depth is:
d(t) = 4 + 2 * cos(2π/6 * (t - 15))
where t is the time in hours after low tide at 15:00 (3:00 PM).