The water depth in Hamburg's harbor basin fluctuates with the tides between 4 and 8 meters. There are almost 12 hours between two highest water depths. The maximum water depth is reached three hours after midnight.
a) Determine the function term of the sine function.
b) Read the period in which the water level is below 5 m within the first 15 hours.
Solution to a) f(x) = 2 Sin(Pi/6(t-3))+6
(1 votes)
Loading...I mean 0 can’t be the answer, can it?
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Yeah, that’s math.
What exactly do you want to know? What’s your problem?
You need to find some things:
The period length of the sine is 2 Pi, you want to stretch it to a period length of 12 .
Then the deepest point should be 3 and you must move the function accordingly.
And then the function should not vary between 1 and −1 but between 4 and 8. This means that the amplitude is not 2 but 4 (by 8-4) and the mean value is not 0 but 6.
The function must be modified for each of these conditions. And if you do, then what has been stated as a solution will come out.
(a)
f(t) indicates the water depth depending on the time (start 0:00).
I come to the following equation:
f(t) = 2 * sin(π / 6) * t) + 6
At 3:00 a.m. the maximum of the water depth is reached:
f(3) = 2 * sin(((π / 6) * 3) + 6 = 2 * sin(π / 2) + 6 = 2 * 1 + 6 = 8
(b)
We calculate the intersections in the interval [0 , 15]
2 * sin((π / 6) * t) + 6 = 5
sin(((π / 6) * t) = -1 / 2
* t = -π / 6 + 2 * π * n ∨ (π / 6) * t = π – (-π / 6) + 2 * π * n , n ε Z
t_1 = -1 + 12 * n = 7 + 12 * n
For n=1, t_1 =11 and for n=0, t_2 =7.
The water depth is below 5 m between 7:00 and 11:00.