Hi,

A fountain in a shopping centre has a single jet of water. The height of the jet of water varies according to a sinusoidal function. A student notes that, in exactly one minute, the jet goes from a minimum height of 2m to a maximum height of 6m and back to 2m.

At 13:00, the jet of water is at a height of 2m.
What will be the height of the jet of water, to the nearest tenth of a metre, when the clock reads 13:12:40? (13hr, 12min, 40sec). Thanks a lot!

F = 1c/min = 1rev/min. = Frquency.

P = 1/F = 1min/rev. = The period.

T = 13:12:40 - 13:0:0 = 12min,40s. =
12 2/3 min. = 38/3 min.

1rev/min * (38/3)min = 12 2/3 rev.
(2/3)rev * 360Deg/rev = 240 Deg.

Ar=240-180 = 60 Deg.=Reference angle.

h = hmax*sinAr = 6*sin60 = 5.2 m.

To find the height of the jet of water at 13:12:40, we can start by understanding the pattern of the jet's height variation over time.

Given that the height of the jet follows a sinusoidal function, we can assume it follows the general form: h(t) = A*sin(B(t - C)) + D, where:
- A represents the amplitude (half the difference between the maximum and minimum heights)
- B represents the frequency (2π divided by the period)
- C represents the phase shift (horizontal displacement)
- D represents the vertical shift (average height)

From the given information, we know that the maximum height is 6m and the minimum height is 2m. Therefore, the amplitude (A) is (6 - 2)/2 = 2m.

To determine the period, we need to find out the time it takes for the jet to go from the minimum height to the maximum height and back to the minimum height. The problem states that this complete cycle takes exactly one minute. Thus, the period (T) is 1 minute or 60 seconds.

Now, we have A = 2 and T = 60 seconds.

Since the time given is 13:12:40 and we want to find the height of the jet at that time, we need to convert that time into seconds. 13 hours, 12 minutes, and 40 seconds can be converted to 13*3600 + 12*60 + 40 = 47560 seconds.

Substitute the values into the equation h(t) = A*sin(B(t - C)) + D:

h(47560) = 2*sin(2π/T * (47560 - C)) + D

The unknown values are B (frequency), C (phase shift), and D (vertical shift). To find them, we need additional information.

If we assume that the jet was at a height of 2m at 13:00, we can use this as a reference point. Since 13:00 is 0 seconds, we can substitute the values into the equation to solve for D:

2 = 2*sin(2π/60 * (0 - C)) + D
2 = 2*sin(-2πC/60) + D

Given that sin(-θ) = -sin(θ), the equation can be simplified to:

2 = -2*sin(2πC/60) + D

Now, we have two unknowns: B and C. We need a second equation to solve for them. This information is crucial but is not provided in the question. Check if there is any other given condition related to the jet's height or plot a graph of the jet's height over time to determine a second equation.

Without additional information, it is not possible to determine the exact height of the jet at 13:12:40.