Question

The depth of water in a tank oscillates sinusoidally once every 6 hours. If the smallest depth is 8.7 feet and the largest depth is 11.3 feet, find a possible formula for the depth in terms of time t in hours. Assume that at t=0 the water level is at the average of the depth and is rising

Answer 1

The formula for the depth of water in the tank as a function of time t is d(t) = 10 + 1.3 sin((π/3) t).

Step-by-step explanation:

To find a possible formula for the depth of water in the tank as a function of time, we can use the equation for a sinusoidal function. The equation is:

d(t) = A + B sin(ωt + φ)

where A is the average depth, B is half the difference between the smallest and largest depth, ω is the angular frequency (2π divided by the period), t is the time, and φ is the phase shift. In this case, the average depth A is the average of the smallest and largest depth, which is 10 feet (8.7 + 11.3) / 2 = 10 feet. The difference between the smallest and largest depth B is half of the difference, which is 1.3 feet (11.3 - 8.7) / 2 = 1.3 feet. The period of the oscillation is 6 hours, so the angular frequency ω is 2π divided by 6 (2π / 6 = π /3). Since the water level is rising at t=0, the phase shift φ is 0. Therefore, the formula for the depth in terms of time t is:

d(t) = 10 + 1.3 sin((π/3) t)