rania is riding the ferris wheel. her vertical height h(t) (in m) off the ground as a function of time t (in…

rania is riding the ferris wheel. her vertical height h(t) (in m) off the ground as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a·cos(b·t)+d. at t = 0, when she starts moving, she is at a height of 10 m off the ground, which is as low as she goes. after 20π seconds, she reaches her maximum height of 30 m. find h(t). t should be in radians. h(t) =

rania is riding the ferris wheel. her vertical height h(t) (in m) off the ground as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a·cos(b·t)+d. at t = 0, when she starts moving, she is at a height of 10 m off the ground, which is as low as she goes. after 20π seconds, she reaches her maximum height of 30 m. find h(t). t should be in radians. h(t) =

Answer

Explanation:

Step1: Find the value of $d$

The mid - height (average of the maximum and minimum) gives $d$. The minimum height is $10$m and the maximum height is $30$m. So, $d=\frac{10 + 30}{2}=20$.

Step2: Find the value of $a$

The amplitude $a$ is the distance from the mid - height to the maximum or minimum. So, $a=30 - 20=10$.

Step3: Find the value of $b$

The period $T$ is the time it takes to go from minimum to maximum and back to minimum. Here, it takes $20\pi$ seconds to go from minimum to maximum. So the period $T = 40\pi$ seconds. The formula for the period of $y = a\cos(bt)+d$ is $T=\frac{2\pi}{b}$. Since $T = 40\pi$, we have $40\pi=\frac{2\pi}{b}$. Solving for $b$ gives $b=\frac{2\pi}{40\pi}=\frac{1}{20}$.

Answer:

$H(t)=10\cos(\frac{1}{20}t)+20$