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.

PLEASE ANSWER ASAP

cos(kt) has period 2pi/k

cos(t) has max at t=0, so -cos(t) has min at t=0

y = 4 - 2cos(pi/6 (t-15))

http://www.wolframalpha.com/input/?i=4+-+2cos%28pi%2F6+%28t-15%29%29+

Why did the water go to therapy? Because it had some low tide issues!

Now, let's tackle your question. We know that one cycle is completed every 12 hours, so the period of the cycle is 12 hours.

When the depth is at its lowest point (low tide), it's 2m. And when it's at the average depth, it's 4m. This means that the amplitude of the function is (4-2)/2 = 1m.

We also know that low tide occurred at 15:00. So we can set up our equation as follows:

d(t) = A * sin(2π / T * (t - t₀)) + D₀

where A is the amplitude, T is the period, t is the time in hours, t₀ is the time of low tide (15:00), and D₀ is the average depth (4m).

Plugging in our values:

d(t) = 1 * sin(2π / 12 * (t - 15)) + 4

And there you have it! The equation for the depth, d(t), as a function of time, t, after low tide occurred at 15:00.

To find an equation for the depth of the water, we need to determine the relationship between the time elapsed since low tide and the depth of the water.

At low tide, the depth is 2m, and at high tide (average depth), the depth is 4m. This means that the water level increases by 2 meters over one cycle (low tide to high tide). Since one cycle is completed every 12 hours, we can say that the water level increases by 2 meters per 12-hour period.

We can use this information to develop an equation for the depth of the water, d(t), with respect to the average depth:

d(t) = 2 + (t / 12) * 2

In this equation, t represents the time elapsed since low tide in hours. The term (t / 12) gives us the number of cycles completed since low tide, and multiplying it by 2 gives us the additional depth gained. Adding this to the initial depth at low tide (2m), we get the final depth at any given time.

So, for example, if we want to find the depth 6 hours after low tide (t = 6), we can substitute this value into the equation:

d(6) = 2 + (6 / 12) * 2
= 2 + 0.5 * 2
= 2 + 1
= 3 meters

Therefore, at 6 hours after low tide, the depth of the water will be 3 meters.

To find an equation of the depth, d(t), with respect to the average depth, as a function of time, t, after low tide, we need to consider the periodic nature of tides.

Given that one cycle is completed approximately every 12 hours, we can use a sinusoidal function to model the depth of the water. The general form of a sinusoidal function is of the form:

f(t) = A * sin(B * (t - C)) + D

Where:
- A: the amplitude of the function (maximum deviation from the average)
- B: the frequency of the function (determines the period)
- C: phase shift (horizontal shift or time offset)
- D: vertical shift (average value of the function)

In this case, we know that the average depth is 4m and the lowest depth (at low tide) is 2m. Therefore:
- Average depth (D) = 4m
- The difference between average and lowest depth (amplitude, A) = 2m

The period of the function (time for one complete cycle) is 12 hours, so the frequency (B) can be calculated using the formula:

B = 2π / P

Where P is the period. Substituting in our given value:

B = 2π / 12 = π / 6

Now, the phase shift (C) can be determined by observing when the low tide occurred. If low tide occurred at 15:00, or 3:00 PM, then the phase shift would be 3 hours ahead. However, we need to convert this time into radians, since the angular argument in the sine function is in radians.

To convert from hours to radians, we can use the fact that 12 hours is equal to 2π radians.

3 hours = (3 / 12) * 2π = π / 2

Therefore, the phase shift (C) is π / 2.

Putting all these values into the equation, we get:

d(t) = 2 * sin((π / 6) * (t - (π / 2))) + 4

So, the equation for the depth, d(t), as a function of time, t, after low tide is d(t) = 2 * sin((π / 6) * (t - (π / 2))) + 4.