derive an equation (d = acos(bt)) for the displacement (d) (in feet) in relation to sea level of a piece of…

derive an equation (d = acos(bt)) for the displacement (d) (in feet) in relation to sea level of a piece of cloth tied to a water wheel over (t) seconds. assume the water wheel is half - submerged and takes 24 seconds to complete one turn. at (t = 0) seconds, the cloth is at a height of 10 feet above sea level, its maximum displacement from the surface of the water. find the values of (a) and (b). (a=) (b=)

derive an equation (d = acos(bt)) for the displacement (d) (in feet) in relation to sea level of a piece of cloth tied to a water wheel over (t) seconds. assume the water wheel is half - submerged and takes 24 seconds to complete one turn. at (t = 0) seconds, the cloth is at a height of 10 feet above sea level, its maximum displacement from the surface of the water. find the values of (a) and (b). (a=) (b=)

Answer

Answer:

$a = 10$, $b=\frac{\pi}{12}$

Explanation:

Step1: Determine the amplitude $a$

The maximum displacement is 10 feet. In the equation $d = a\cos(bt)$, the amplitude $a$ is the maximum value of $d$. So $a = 10$.

Step2: Find the value of $b$

The period $T$ of the cosine - function is related to $b$ by the formula $T=\frac{2\pi}{b}$. The water - wheel takes 24 seconds to complete one turn, so the period $T = 24$ seconds. Substituting $T = 24$ into $T=\frac{2\pi}{b}$, we get $24=\frac{2\pi}{b}$. Solving for $b$: [ \begin{align*} b&=\frac{2\pi}{24}\ b&=\frac{\pi}{12} \end{align*} ]