The curvature at a point P of a parametric curve x = x(t), y = y(t) is given below, where the dots indicate derivatives with respect to t, so ẋ = dx/dt. Use the formula to find the curvature of the cycloid x = θ − sin(θ), y = 1 − cos(θ) at the top of one of its arches. $ \kappa = \dfrac {\left| \dot{x} \ddot{y} - \ddot{x} \dot{y} \right|} {[\dot{x}^2 + \dot{y}^2]^{3/2}} $

In one day there are too high tides into low tides and equally spaced intervals the high tide is observed to be 6 feet above the average sea level after six hours passed a low tide occurs at 6 feet below the average sea level in this task you will model this occurrence using a trigonometric function by using x as a measurement of time assume the first high tide occurs at x=0. a. what are the independent and dependent variables? b. determine these key features of the function that models the tide: 1.amplitude 2.period 3.frequency 4.midline 5.vertical shift 6.phase shift c. create a trigonometric function that models the ocean tide for a period of 12 hours. d. what is the height of the tide after 93 hours?