The average depth of the water in a port on a tidal river is 4m. At low tide, the depth of the water is 2m. One cycle is completed approximately every 12h.
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.
To find the equation of the depth, d(t) with respect to the average depth, we need to take into account that the depth fluctuates due to tides.
Let's define the start of the cycle as the time of low tide, which occurred at 15:00. At low tide, the depth is 2m, and at high tide (or average depth), the depth is 4m.
Since one cycle is completed approximately every 12 hours, we can divide the 12-hour cycle into two equal halves: the rising tide and the falling tide. Each half-cycle takes 6 hours.
During the rising tide, the depth increases from 2m to 4m, and during the falling tide, it decreases from 4m back to 2m. These changes occur symmetrically around the average depth.
To find the equation of the depth, we can use a sinusoidal function, specifically a sine function, since it exhibits the desired symmetry around the average depth.
Let's define the time t=0 as the start of the rising tide, which occurs 6 hours after low tide at 21:00. We can write the equation:
d(t) = A * sin(B(t - C)) + D
Where:
A is the amplitude (half the difference between high and low tide) = (4-2)/2 = 1
B is the frequency (2π divided by the length of the cycle) = 2π/12 = π/6
C is the horizontal shift (time elapsed since the start of the rising tide) = t - 6
D is the vertical shift (average depth) = 4
Substituting these values into the equation, we get:
d(t) = 1 * sin((π/6)(t - 6)) + 4
Therefore, the equation of the depth, d(t), as a function of time, t hours, after low tide is given by:
d(t) = sin((π/6)(t - 6)) + 4