Answer:
(a) The time to complete 1 cycle and return is 16/3
(b) The minimum height is 30 inches and the maximum is 54 inches
Step-by-step explanation:
Given
[tex]f(t)=12\sin(\frac{3\pi}{8}t) + 42[/tex]
Solving (a): Time to complete 1 cycle and return
This implies that we calculate the period.
This is calculated using:
[tex]T = \frac{2\pi}{w}[/tex]
Where:
[tex]w = \frac{3\pi}{8}[/tex]
So, we have:
[tex]T = \frac{2\pi}{3\pi/8}[/tex]
Rewrite as:
[tex]T = \frac{8*2\pi}{3\pi}[/tex]
[tex]T = \frac{16}{3}[/tex]
Solving (b): The maximum and the minimum height
To do this, we have:
[tex]-1 \le \sin(\theta) \le 1[/tex]
Which means:
[tex]-1 \le \sin(\frac{3\pi}{8}) \le 1[/tex]
So, the maximum and the minimum of [tex]\sin(\frac{3\pi}{8})[/tex] are:
[tex]\sin(\frac{3\pi}{8}) = -1[/tex] --- minimum
[tex]\sin(\frac{3\pi}{8}) = 1[/tex] --- maximum
Given that:
[tex]f(t)=12\sin(\frac{3\pi}{8}t) + 42[/tex]
So, the minimum height is:
[tex]h_{min} = 12 * -1 + 42 = 30[/tex]
The maximum is:
[tex]h_{max} = 12 * 1 + 42 = 54[/tex]
