4. the functions f and g are given by f(x) = cos(x) and g(x) = cos(5x). how are the graphs of f and g…

Explanation:

Step1: Analyze cos - function transformation

The general form of a cosine - function is $y = A\cos(Bx - C)+D$. For $f(x)=\cos(x)$ and $g(x)=\cos(5x)$, $A = 1$, $C = 0$, $D = 0$ for both functions. The only difference is $B$. The period of $y=\cos(x)$ is $T_1 = 2\pi$, and the period of $y=\cos(5x)$ is $T_2=\frac{2\pi}{5}$. The graph of $g(x)=\cos(5x)$ is a horizontal compression of the graph of $f(x)=\cos(x)$ by a factor of $\frac{1}{5}$.

Step2: Set up height - equation for windmill

The height of a point on a rotating object can be modeled using a cosine or sine function. The center of the wind - mill is at a height of $h_0 = 6$ feet. The length of the blade is $r = 1.5$ feet. We know that the vertical displacement of the point $P$ from the center of the wind - mill is given by $y=r\sin(\theta)$, where $\theta$ is the angle of rotation. Here, the initial angle is $\frac{\pi}{4}$, so the total angle of rotation is $\theta=a+\frac{\pi}{4}$. The height $h$ of the point $P$ is the sum of the height of the center of the wind - mill and the vertical displacement of the point $P$ from the center of the wind - mill. So, $h = 6+1.5\sin(a + \frac{\pi}{4})$.

Answer:

4. The graph of $g(x)=\cos(5x)$ is a horizontal compression of the graph of $f(x)=\cos(x)$ by a factor of $\frac{1}{5}$.
5. $h = 6+1.5\sin(a+\frac{\pi}{4})$