Rania is riding the ferris wheel. Her vertical height H(t)H(t)H, left parenthesis, t, right parenthesis (in \text{m}mstart text, m, end text) off the ground as a function of time ttt (in seconds) can be modeled by a sinusoidal expression of the form a\cdot\cos(b\cdot t)+da⋅cos(b⋅t)+da, dot, cosine, left parenthesis, b, dot, t, right parenthesis, plus, d. At t=0t=0t, equals, 0, when she starts moving, she is at a height of 10\text{ m}10 m10, start text, space, m, end text off the ground, which is as low as she goes. After 20\pi20π20, pi seconds, she reaches her maximum height of 30\text{ m}30 m30, start text, space, m, end text. Find H(t)H(t)H, left parenthesis, t, right parenthesis.

The amplitude of the cosine function is half that, or 10 m. Here, the function starts off at a minimum, whereas the parent cosine function starts at a maximum. That means the parent function is reflected vertically, so the amplitude coefficient in H(t) is negative.

The midline of the cosine function is the average of the extreme heights, so is ...

midline = (30 m +10 m)/2 = 20 m

The period of the function is double the time it takes to go from a minimum to a maximum, so is ...

p = 2(20π) = 40π . . . . seconds

In the given function form ...

H(t) = a·cos(bt) +d

the amplitude is a=-10; the coefficient b is 2π/p = 1/20; and the midline is d = 20. Then the function is ...

H(t) = -10·cos(t/20) +20