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!
To determine the height of the jet of water at 13:12:40, we need to calculate the time elapsed since 13:00 and use it to find the corresponding height of the sinusoidal function.
First, we need to find out the time elapsed in minutes from 13:00 to 13:12:40.
Elapsed time in minutes = (hours * 60) + minutes + (seconds / 60)
Elapsed time in minutes = (13 * 60) + 12 + (40 / 60)
Elapsed time in minutes = 780 + 12 + (40 / 60)
Elapsed time in minutes = 792 + (2/3)
Now, we need to determine the period of the sinusoidal function. The period of a sinusoidal function is the time taken to complete one full cycle.
Given that the jet of water goes from a minimum height of 2m to a maximum height of 6m and back to 2m in exactly one minute, we can conclude that the period of the sinusoidal function is one minute.
Since the period is one minute, the height of the jet of water repeats every minute.
Now, we can find the height of the jet of water at 13:12:40 by finding the remainder when dividing the elapsed time by the period and using this remainder to find the height.
Elapsed time in minutes modulo period = elapsed time in minutes % period
Elapsed time in minutes modulo period = 792 + (2/3) % 1
Elapsed time in minutes modulo period = (3 * 792 + 2) % 3 / 3
Elapsed time in minutes modulo period = 2378 % 3 / 3
Elapsed time in minutes modulo period = 2 / 3
Therefore, the height of the jet of water at 13:12:40 will be the same as the height at 13:00, which is 2m.
So, the height of the jet of water at 13:12:40 is approximately 2.0 meters.
To find the height of the jet of water at 13:12:40, we need to determine the time elapsed from 13:00 to 13:12:40.
First, let's find the period of the sinusoidal function. The given information states that the jet of water takes exactly one minute to go from the minimum height of 2m to the maximum height of 6m and back to 2m.
Since the complete oscillation takes one minute, the period of the sinusoidal function is one minute.
Next, let's determine the angular frequency (ω) of the sinusoidal function. The angular frequency is given by:
ω = 2π / period
In this case, the period is one minute, so:
ω = 2π / 1 = 2π rad/min
Now, we can write the equation for the height of the jet of water as a function of time:
h(t) = A * sin(ωt) + h0
Where:
- h(t) is the height of the jet of water at time t
- A is the amplitude of the oscillation (half the difference between the minimum and maximum heights)
- ω is the angular frequency (2π rad/min)
- t is the time in minutes
- h0 is the initial height of the jet of water at 13:00 (2m)
We are given that at 13:00, the jet of water is at a height of 2m, so h0 = 2m.
Now, let's find A, the amplitude of the oscillation:
A = (maximum height - minimum height) / 2 = (6m - 2m) / 2 = 2m
So, the equation for the height of the jet of water becomes:
h(t) = 2 * sin(2πt) + 2
To find the height of the jet of water at 13:12:40, we need to convert 13:12:40 to minutes.
13:12:40 = 13 * 60 + 12 + 40/60 = 792 + 12 + 2/3 = 804 + 2/3 = 804.667 minutes
Substituting t = 804.667 into the equation h(t):
h(804.667) = 2 * sin(2π * 804.667) + 2
Calculating this expression will give us the height of the jet of water at 13:12:40, rounded to the nearest tenth of a meter.