May someone help me with this HW problem?
The depth (D metres) of water in a harbour at a time (t hours) after midnight on a particular day can be modelled by the function
D = 2 sin(0.51t - 0.4) +5, t <=15,
where radians have been used.
Select the two options which are correct statements about the predictions based on this model.
Select one or more:
a) The smallest depth is 5 metres.
b) At midday, the depth is approximately 7 metres.
c) The model can be used to predict the tide for up to 15 days.
d) The largest depth is 7 metres.
e) At midnight the depth is approximately 4.2 metres.
f) The time between the two high tides is exactly 12 hours.
g) The depth of water in the harbour falls after midnight.
not a, shallow = -2+5 = 3
check b, 2 sin (.51*12 -.4) + 5 = 2 sin (around 2 pi) +5 around 5, no
t is HOURS </= 15, not days] so not c
YES d, because 2+5 = 7
e. when t = 0 D = 2 sin (-.4 ) + 5 = - 2(.4)+5 = 5-.8 =4.2 YES
f. .51*12 = which is close but not exactly 2 pi. In fact nothing but 2 pi is exactly 2 pi
g. well, it goes up and down
a) The smallest depth is 5 metres.
b) At midday, the depth is approximately 7 metres.
Explanation:
a) The equation gives a constant value of 5 for D when t <= 15, indicating that the smallest depth is 5 meters.
b) At midday, which is 12 hours after midnight, t = 12. By plugging this value into the equation, we get D = 2 sin(0.51(12) - 0.4) + 5 ≈ 7. Therefore, the depth at midday is approximately 7 meters.
The other options are incorrect:
c) The model can only be used to predict the tide up to 15 hours after midnight, not days.
d) The equation doesn't give a constant value for the largest depth. It varies depending on the time.
e) At midnight, t = 0. By plugging this into the equation, we get D = 2 sin(0.51(0) - 0.4) + 5 ≈ 4.2. Therefore, the depth at midnight is approximately 4.2 meters.
f) The equation does not give a fixed time interval between two high tides. It depends on the values of t at which the sine function reaches its maximum and minimum.
g) The depth of water in the harbor can rise or fall depending on the values of t. It is not guaranteed to fall after midnight.
To answer this question, let's analyze each statement one by one:
a) The smallest depth is 5 meters.
From the given equation: D = 2 sin(0.51t - 0.4) + 5
The minimum value of the sine function is -1, so the smallest possible value of D will be:
D = 2 (-1) + 5 = 3
Therefore, statement a) is not correct. The smallest depth is 3 meters, not 5 meters.
b) At midday, the depth is approximately 7 meters.
To find the depth at midday, we substitute t = 12 into the equation:
D = 2 sin(0.51(12) - 0.4) + 5
D ≈ 2 sin(6.12 - 0.4) + 5
D ≈ 2 sin(5.72) + 5
D ≈ 2(0.994) + 5
D ≈ 6.988
Therefore, statement b) is correct. At midday, the depth is approximately 7 meters.
c) The model can be used to predict the tide for up to 15 days.
The given equation is only valid when t ≤ 15. Therefore, the model can only be used to predict the tide for up to 15 hours, not days. Therefore, statement c) is not correct.
d) The largest depth is 7 meters.
The maximum value of the sine function is 1, so the largest possible value of D will be:
D = 2(1) + 5 = 7
Therefore, statement d) is correct. The largest depth is 7 meters.
e) At midnight, the depth is approximately 4.2 meters.
To find the depth at midnight, we substitute t = 0 into the equation:
D = 2 sin(0.51(0) - 0.4) + 5
D ≈ 2 sin(-0.4) + 5
D ≈ 2(-0.389) + 5
D ≈ 4.222
Therefore, statement e) is correct. At midnight, the depth is approximately 4.2 meters.
f) The time between the two high tides is exactly 12 hours.
In the given equation, the period of the sine function is 2π/0.51 ≈ 12.36. This means the time between two high tides is approximately 12.36 hours, not exactly 12 hours. Therefore, statement f) is not correct.
g) The depth of water in the harbor falls after midnight.
As we saw in statement e), the depth at midnight is approximately 4.2 meters. Since the given equation represents a sinusoidal function, the depth will oscillate between its maximum and minimum values. Therefore, the depth of water in the harbor will rise and fall after midnight. Therefore, statement g) is not correct.
In summary, the correct statements are:
b) At midday, the depth is approximately 7 meters.
d) The largest depth is 7 meters.
e) At midnight, the depth is approximately 4.2 meters.
To solve this problem, we can analyze the given function and use it to make predictions.
The given function is: D = 2 sin(0.51t - 0.4) + 5, where t represents the time in hours after midnight.
a) The smallest depth is 5 meters:
This statement is correct. The "+5" term in the function means that the minimum value for D is 5, so the smallest depth is indeed 5 meters.
b) At midday, the depth is approximately 7 meters:
To find the depth at midday (12 hours after midnight), we substitute t = 12 into the equation:
D = 2 sin(0.51(12) - 0.4) + 5
Calculating this expression will give us the approximate depth at midday. So we can't determine if this statement is true without further computation.
c) The model can be used to predict the tide for up to 15 days:
This statement is incorrect. The given function is only valid for t ≤ 15 hours (or one day), as indicated by the condition t ≤ 15 in the question. Therefore, the model cannot be used to predict the tide for up to 15 days.
d) The largest depth is 7 meters:
This statement is incorrect. The maximum value of the function can be found by evaluating the function at various t values or by analyzing the sine function. The maximum value of the sine function is 1, so the maximum depth of water is 2(1) + 5 = 7 meters.
e) At midnight, the depth is approximately 4.2 meters:
To find the depth at midnight (t = 0), we substitute t = 0 into the equation:
D = 2 sin(0.51(0) - 0.4) + 5
Calculating this expression will give us the approximate depth at midnight. So we can't determine if this statement is true without further computation.
f) The time between the two high tides is exactly 12 hours:
This statement is incorrect. The function does not have a period of exactly 12 hours, as indicated by the coefficient of t: 0.51. The period can be found by calculating the period of the sine function, but it will not be exactly 12 hours.
g) The depth of water in the harbor falls after midnight:
This statement is true. From the given equation, we can see that the coefficient of the sine function (0.51t - 0.4) is positive, which means that the argument of the sine function is increasing with time. Therefore, the depth of water in the harbor decreases after midnight.
In summary, the correct statements are:
a) The smallest depth is 5 meters.
d) The largest depth is 7 meters.
g) The depth of water in the harbor falls after midnight.