During high tide the water depth in a harbour is 22 m and during low tide it is 10m(Assume a 12h cycle).
Calculate the times at which the water level is at 18m during the first 24 hours.
My solution:
first I found the cos equation: H(t)=-6cos(π/6t)+16
then..
Let π/6t=Θ
18=-6cosΘ+16
18-16=-6cosΘ
Θ=1.230959417
Then I don't know what's next....
H(t)=-6\cos(\frac{\pi}{6} t)+16
cosΘ < 0 in QII and QIII
arccos(-1/3) = 1.91 = pi-1.23
or pi+1.23 = 4.37
So, t = (6/pi) * (1.91 or 4.37)
t = 3.64 or 8.35 hours
To find the times at which the water level is at 18m during the first 24 hours, you need to solve for t in the equation Θ = π/6t.
1. Convert Θ back to radians:
Θ = 1.230959417
2. Solve for t:
π/6t = 1.230959417
To isolate t, divide both sides of the equation by π/6:
t = 1.230959417 * 6/π
3. Calculate t:
t ≈ 2.46804465
Since t represents the number of hours after high tide, you can add this value to the time of high tide to find the times at which the water level is at 18m during the first 24 hours.
For example:
High tide: 12:00 PM
t = 2.46804465 hours ≈ 2 hours and 28 minutes
So the first time the water level reaches 18m is approximately 2 hours and 28 minutes after high tide (around 2:28 PM).
To find the other times, you can add multiples of the tidal cycle (12 hours) to the initial time. However, given the information provided, we are unable to determine additional occurrences within the 24-hour period.
To calculate the times at which the water level is at 18m during the first 24 hours, you have already found the value of Θ to be approximately 1.230959417. Now, let's complete the calculation and convert Θ back to time.
You have the equation: Θ = π/6 * t
Substituting the value of Θ = 1.230959417, we get:
1.230959417 = (π/6) * t
To solve for t, we can rearrange the equation:
t = (1.230959417 * 6) / π
Using the approximation π ≈ 3.14159, we can calculate:
t ≈ (1.230959417 * 6) / 3.14159
t ≈ 2.355190347
So, the value of t is approximately 2.355190347. To convert this value to hours, we can multiply it by the length of the tide cycle, which is 12 hours:
Time = 2.355190347 * 12
Time ≈ 28.26228416
Therefore, the water level reaches 18m approximately 28.26 hours from the start of the tide cycle.
To find the times within the first 24 hours when the water level is at 18m, we need to take into account that the tide cycle repeats after 12 hours.
Since 28.26 hours is larger than 24 hours, we need to find the remainder when dividing 28.26 by 24 to find the time within the first 24 hours:
Remainder = 28.26 mod 24
Remainder ≈ 4.26
Therefore, the water level reaches 18m within the first 24 hours approximately at 4.26 hours after the start of the tide cycle.