Question

The height of the saddle off horse above the base ofa carousel can be modeled 4t by the equation f-rr) : 12 sin ^ r 42, where I represents seconds after the ride started. I How much time does to take for the horse to complete one cycle of motion and return to its starting height. What is the maximum height and the minimum height of the horse's saddle above the base ofthe carousel

Answer 1

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

$f(t) = 12\sin(\frac{3\pi}{8}t) + 42$

Solving (a): Time to complete 1 cycle and return

This implies that we calculate the period.  This is calculated using:

$T = \frac{2\pi}{w}$

Where:

$w =\frac{3\pi}{8}$

So, we have:

$T = \frac{2\pi}{\frac{3\pi}{8}}$

$T = \frac{2}{\frac{3}{8}}$

$T = \frac{2*8}{3}$

$T = \frac{16}{3}$

Solving (b): The maximum and the minimum height

To do this, we have:

$-1 \le \sin(\theta) \le 1$

Which means:

$-1 \le \sin(\frac{3\pi}{8}) \le 1$

So, the minimum is:

$\sin(\frac{3\pi}{8}) =- 1$

And the maximum is:

$\sin(\frac{3\pi}{8}) =1$

Recall that the height is:

$f(t) = 12\sin(\frac{3\pi}{8}t) + 42$

So, the maximum and the minimum of  are:

$h_{min} =12 * -1 + 42$

$h_{min} =30$

and

$h_{max} =12*1+42$

$h_{max} =54$