The depth d of water in a tank oscillates sinusoidally once every
4 hours. If the smallest depth is 7.9 feet and the largest depth is 10.1 feet, find a possible formula for the depth in terms of time t in hours. (Let the water depth be at the average when
t = 0.)
I got y=1.1cos(1/2pi t)
amplitude: (10.1-7.9)/2
B=2pi/4 = 1/2pi
No C because the water depth is at the midline when t=0?
but this was marked wrong
The depth at midline is 7.9+1.1 = 9.0
f(t) is at midline at t=0, so you want a sine function, not cosine, which starts at a max.
So, f(t) = 9.0 ± 1.1 sin(π/2 t)
I use ± because they don't say whether the water is rising or falling at t=0.
Thank you so much! That helped a lot!
Your understanding is correct, except for a small mistake in the formula. The correct formula for the depth in terms of time is:
y = 0.6cos(π/2t)
Let's break it down step by step:
1. The amplitude is the distance from the midline to the maximum or minimum value. In this case, it would be half the difference between the smallest and largest depths:
Amplitude = (10.1 - 7.9) / 2 = 2.2 / 2 = 1.1
2. The period is the time it takes for one complete oscillation. Given that it oscillates once every 4 hours, the period would be 4 hours. Using the formula T = 2π / |B|, where B is the coefficient of t in the formula, we can find the coefficient:
4 = 2π / |B|
|B| = 2π / 4 = π / 2
3. Since the water depth is at the midline when t = 0, there is no need for a constant term (C) in the formula.
Combining these results, the formula for the depth in terms of time is:
y = 1.1cos(π/2t)
So, the correct formula should be y = 1.1cos(π/2t).
Your approach is correct, but there is a small mistake in your formula. The correct formula should be:
y = 1.1 cos((2π/4)t)
Let's break down the formula to understand why:
1. Amplitude (A): The amplitude is the distance from the midline (average depth) to the highest or lowest point. In this case, the smallest depth is 7.9 feet, and the largest depth is 10.1 feet. So, the amplitude is (10.1 - 7.9)/2 = 1.1 feet.
2. Period (T): The period is the time it takes for one complete cycle of oscillation. In this case, the depth oscillates once every 4 hours, so the period is 4 hours.
3. Angular frequency (ω): The angular frequency is related to the period and is given by 2π divided by the period (T). In this case, ω = 2π/4 = 1/2π.
Therefore, the formula for the depth in terms of time (t) in hours is:
y = 1.1 cos((1/2π)t)
Make sure to double-check your calculations and adjust your formula accordingly.