the reflector is attached to the wheel of a mountain bike. the height, $h$, in inches, of the reflector with…

the reflector is attached to the wheel of a mountain bike. the height, $h$, in inches, of the reflector with respect to the ground can be modeled by the function $h(t)=13 + 9.5cos(\frac{40pi}{3}t)$, where $t$ is the time, in seconds. when the function is graphed on a coordinate plane, which of the following interpretations are correct? select three that apply.\na the mid - line of the height of the reflector is 9.5 inches.\nb the mid - line of the height of the reflector is 13.0 inches.\nc the maximum height reached by the reflector is 22.5 inches.\nd the period indicates that the reflector makes 0.15 of a complete turn per second.\ne the period indicates that the reflector makes a complete turn every 0.15 seconds.

the reflector is attached to the wheel of a mountain bike. the height, $h$, in inches, of the reflector with respect to the ground can be modeled by the function $h(t)=13 + 9.5cos(\frac{40pi}{3}t)$, where $t$ is the time, in seconds. when the function is graphed on a coordinate plane, which of the following interpretations are correct? select three that apply.\na the mid - line of the height of the reflector is 9.5 inches.\nb the mid - line of the height of the reflector is 13.0 inches.\nc the maximum height reached by the reflector is 22.5 inches.\nd the period indicates that the reflector makes 0.15 of a complete turn per second.\ne the period indicates that the reflector makes a complete turn every 0.15 seconds.

Answer

Explanation:

Step1: Recall the general form of a cosine - function

The general form of a cosine - function is $y = A\cos(Bx - C)+D$, where $D$ is the mid - line, $A$ is the amplitude, and the period $T=\frac{2\pi}{|B|}$. For the function $h(t)=13 + 9.5\cos(\frac{40\pi}{3}t)$, we have $A = 9.5$, $B=\frac{40\pi}{3}$, and $D = 13$.

Step2: Find the mid - line

The mid - line of the cosine function $h(t)$ is given by the value of $D$. So, the mid - line is $y = 13$. Option A is incorrect and option B is correct.

Step3: Find the maximum value

The maximum value of a cosine function $y = A\cos(Bx - C)+D$ is $D + A$. Substituting $A = 9.5$ and $D = 13$, we get $13+9.5=22.5$. So, the maximum height reached by the reflector is 22.5 inches. Option C is correct.

Step4: Find the period

The period $T$ of the function $y=\cos(Bt)$ is $T=\frac{2\pi}{|B|}$. For $h(t)=13 + 9.5\cos(\frac{40\pi}{3}t)$, $B=\frac{40\pi}{3}$, so $T=\frac{2\pi}{\frac{40\pi}{3}}=\frac{2\pi\times3}{40\pi}=0.15$ seconds. This means the reflector makes a complete turn every 0.15 seconds. Option E is correct and option D is incorrect.

Answer:

B. The mid - line of the height of the reflector is 13.0 inches. C. The maximum height reached by the reflector is 22.5 inches. E. The period indicates that the reflector makes a complete turn every 0.15 seconds.