On a typical day at an ocean port, the water has a maximum depth of 18m at 6:00 am. The minimum depth of 9m occurs 6.8 hours later. Write an equation to describe the relationship between the depth and time.

the period is 2*6.8 = 13.6, so we have

d(t) = a*cos(2pi/13.6 t) = a*cos(.462t)

The max-min range is 18-9 = 9, so the amplitude is 4.5 and we have

d(t) = 4.5 cos(.462t)

If we call t is the hour of the day, then max is at t=6, so

d(t) = 4.5 cos(.462(t-6))

Since the average depth is (9+18)/2, we have

d(t) = 4.5 cos(.462(t-6)) + 13.5

To see the graph, visit

http://rechneronline.de/function-graphs/

enter the function as

4.5 cos(.462*(x-6))+13.5

and set the x-range at 0-20
the y-range at 0-20

To describe the relationship between the depth and time at the ocean port, we can use a linear equation since the change in depth over time is constant.

Let's denote the time (in hours) as 't' and the depth (in meters) as 'd'.

Given that the maximum depth of 18m occurs at 6:00 am (t=0) and the minimum depth of 9m occurs 6.8 hours later, we can set up the equation as follows:

d = mt + c

where m is the rate of change (slope) and c is the y-intercept.

Now, let's find the slope (m). The change in depth is from 18m to 9m over a time period of 6.8 hours. So, the rate of change is:

m = (9 - 18) / 6.8
m = -9 / 6.8
m ≈ -1.32

The negative sign indicates that the depth is decreasing over time.

Next, we can substitute the slope and one of the known points (6:00 am: depth = 18m, t = 0) into the equation to find the y-intercept (c).

18 = -1.32(0) + c
18 = c

Therefore, the equation that describes the relationship between the depth (d) and time (t) is:

d = -1.32t + 18.

To write an equation describing the relationship between the depth and time at the ocean port, we can use the information given: the maximum depth of 18m at 6:00 am and the minimum depth of 9m occurring 6.8 hours later.

Let's assign time t=0 to represent 6:00 am. Since the minimum depth occurs 6.8 hours later, we can represent that as t=6.8.

We know that as time increases, the depth decreases. Therefore, there is a linear relationship between depth and time.

Now, we can set up an equation with the information given. The general equation for a linear relationship is y = mx + b, where y represents the depth and x represents time. We need to determine the values of m (slope) and b (y-intercept).

At t=0, the depth is 18m, so we have the point (0, 18). Plugging this into the equation, we get:
18 = m(0) + b
18 = b

Now, at t=6.8, the depth is 9m, so we have the point (6.8, 9). Plugging this into the equation, we get:
9 = m(6.8) + 18

Simplifying, we have:
0.2m = -9
m = -9 / 0.2
m = -45

Therefore, the equation describing the relationship between depth and time is:
y = -45x + 18

In this equation, y represents the depth in meters, and x represents time in hours, with t=0 corresponding to 6:00 am.