Tides are cyclical phenomena caused by the gravitational pull of the sun and the moon. On a particular retaining wall, the ocean generally reaches the 3m mark at high tide. At low tide, the water reaches the 1m mark. Assume that high tide occurs at 12:00 pm and at 12:00 am, and that low tide occurs at 6:00pm and 6:00am. What is the height of the water at 10:30 am?

First problem:

The height of tides follows a sinusoidal path, that is , their height is represented a either a sine or cosine curve, there is no constant change.
Second problem: what is the difference between 12:00 pm and 12:00 am ? Both are midnight.

http://askascientist.co.uk/physics/day-midnight-belong/

(that is why countries that use the metric system also use the 24:00 hour method of measuring time)

I will assume:
0:00 ---> 3 m
6:00 ---> 1 m
12:00 ---> 3 m
18:00 ---> 1 m
24:00 ---> 3 m

So according to your data, a tide period is 12 hours, (not really but ....)
and the amplitude of the cosine curve is 1 m
2π/k = 12
k = π/6

my function is:
height = cos (π/6)t + 2 , where t is the hours after midnight.

testing:
midnight --> t =0 , height = cos 0 + 2 = 1+2 = 3
6:00 -----> t = 6, height = cosπ + 2 = -1+2 = 1
12:00 (noon) --> t = 12, height = cos2π + 2 = 3
looks good

so you want the tide at 10:30 am or t = 10.5
height = cos π(10.5)/6 + 2
= appr 2.71 m

check:
http://www.wolframalpha.com/input/?i=plot+y+%3D+cos+(%CF%80t%2F6)+%2B+2

So this would mean that there is 6 hours in each cycle. 10:30am is 4.5 hours into the rising cycle out of 6 total hours, or 4.5/6 . This ratio is directly proportional to the tide marking assuming that this is a constant change. The change in height from 3 to 1 is 2m. Thus we should multiply our ratio by 2m. 2*(4.5/6)=4.5/3=1.5. So the tide is at 1m+1.5m=2.5m by 10:30 am.

To determine the height of the water at 10:30 am, we need to analyze the tide cycle.

It is given that high tide occurs at 12:00 pm and 12:00 am, while low tide occurs at 6:00 pm and 6:00 am. This means that there are 6 hours between each high tide and low tide.

Let's break down the time intervals:

12:00 am to 6:00 am = 6 hours (low tide to high tide)
6:00 am to 12:00 pm = 6 hours (high tide to low tide)
12:00 pm to 6:00 pm = 6 hours (low tide to high tide)

As we can see, 10:30 am falls within the time period from high tide to low tide. More specifically, it is 4.5 hours after the high tide at 12:00 pm.

Now, to calculate the change in tide, we can compare the difference between the high tide and low tide measurements:

High Tide (3m) - Low Tide (1m) = Change in Tide (2m)

Since it is 4.5 hours after 12:00 pm, we can determine the proportional change in tide:

4.5 hours / 6 hours = 0.75

To find the height of the water at 10:30 am, we need to determine how much of the change in tide occurred in the time interval from high tide to 10:30 am:

Change in Tide (2m) x 0.75 = 1.5m

Therefore, the height of the water at 10:30 am would be 1.5 meters.

To determine the height of the water at 10:30 am, we need to understand the tidal cycle and use the given information.

Tidal cycles are generally divided into two high tides and two low tides per day, with each cycle lasting approximately 12 hours and 25 minutes. High tides occur when the water level reaches its peak, while low tides occur when the water level is at its lowest.

Given that high tide occurs at 12:00 pm and 12:00 am, and low tide occurs at 6:00 pm and 6:00 am, we can deduce the following tidal pattern:

12:00 pm - High Tide
6:00 pm - Low Tide
12:00 am - High Tide
6:00 am - Low Tide

Now, let's determine the height of the water at 10:30 am using this pattern.

From 12:00 am to 12:00 pm, six and a half hours have passed. Since a complete tidal cycle takes approximately 12 hours and 25 minutes, we can assume that a little over half of a tidal cycle has occurred.

Given that the water reaches the 3m mark at high tide and the 1m mark at low tide, we can assume that the water level has risen above the 1m mark. However, since we are closer to low tide than high tide, the water level would be closer to the low tide mark.

So, the height of the water at 10:30 am would be around 1m or slightly higher, based on the given information and the tidal cycle.

It's important to note that tides can be affected by various other factors, such as the shape of the coastline and local weather conditions. This explanation assumes a simplified tidal pattern and does not account for any exceptional circumstances that may affect the specific location's tides.