use the drawing tools to form the correct answer on the graph. a buoy marks a channel for boat navigation…

use the drawing tools to form the correct answer on the graph. a buoy marks a channel for boat navigation and bobs up and down with the motion of the waves. in the given function, $f(t)$ represents the height of the buoy above sea level, in feet, after $t$ seconds. $f(t)=2cos(\frac{pi}{4}t)$ on the graph, plot the points where the height, $f(t)$, is at a minimum.

use the drawing tools to form the correct answer on the graph. a buoy marks a channel for boat navigation and bobs up and down with the motion of the waves. in the given function, $f(t)$ represents the height of the buoy above sea level, in feet, after $t$ seconds. $f(t)=2cos(\frac{pi}{4}t)$ on the graph, plot the points where the height, $f(t)$, is at a minimum.

Answer

Answer:

The minimum value of (y = f(t)=2\cos(\frac{\pi}{4}t)) is (- 2). We know that (\cos x) has a minimum value of (-1). For (y = 2\cos(\frac{\pi}{4}t)), when (\cos(\frac{\pi}{4}t)=-1), (\frac{\pi}{4}t=(2n + 1)\pi), (n\in\mathbb{Z}). Solving for (t): [ \begin{align*} \frac{\pi}{4}t&=(2n + 1)\pi\ t&=4(2n + 1)\ t&=8n+4,n\in\mathbb{Z} \end{align*} ] In the domain of the given graph ((0\leq t\leq16)), when (n = 0), (t = 4); when (n = 1), (t=12). So the points to be plotted are ((4,-2)) and ((12,-2))

Explanation:

Step1: Recall cosine - function property

The minimum of (\cos x=-1)

Step2: Set up equation for minimum of given function

Set (\cos(\frac{\pi}{4}t)=-1)

Step3: Solve for (t)

(\frac{\pi}{4}t=(2n + 1)\pi), then (t = 8n + 4)

Step4: Find (t) values in given domain

For (n = 0,t = 4); for (n = 1,t = 12)