Alright, let's tackle this problem step-by-step.
### Part I: Object's position at [tex]\( t = 0 \)[/tex]
The equation for the object's displacement is given by:
[tex]\[ d = 3 \sin(t) - 2 \][/tex]
To find the position at [tex]\( t = 0 \)[/tex], substitute [tex]\( t = 0 \)[/tex] into the equation:
[tex]\[ d = 3 \sin(0) - 2 \][/tex]
Since [tex]\(\sin(0) = 0\)[/tex]:
[tex]\[ d = 3 \cdot 0 - 2 \][/tex]
[tex]\[ d = -2 \][/tex]
Thus, the object's position at [tex]\( t = 0 \)[/tex] is:
[tex]\[ \boxed{-2 \text{ meters}} \][/tex]
### Part II: Maximum displacement from the [tex]\( t = 0 \)[/tex] position
The sine function, [tex]\(\sin(t)\)[/tex], oscillates between -1 and 1.
1. Maximum displacement:
For the maximum value of [tex]\(\sin(t) = 1\)[/tex]:
[tex]\[ d_{\text{max}} = 3 \sin(1) - 2 \][/tex]
[tex]\[ d_{\text{max}} = 3 \cdot 1 - 2 \][/tex]
[tex]\[ d_{\text{max}} = 1 \][/tex]
2. Minimum displacement:
For the minimum value of [tex]\(\sin(t) = -1\)[/tex]:
[tex]\[ d_{\text{min}} = 3 \sin(-1) - 2 \][/tex]
[tex]\[ d_{\text{min}} = 3 \cdot (-1) - 2 \][/tex]
[tex]\[ d_{\text{min}} = -5 \][/tex]
Therefore, the maximum and minimum displacements from the [tex]\( t = 0 \)[/tex] position are:
[tex]\[ \boxed{1 \text{ meter}} \ \text{(maximum displacement)} \][/tex]
[tex]\[ \boxed{-5 \text{ meters}} \ \text{(minimum displacement)} \][/tex]
### Part III: Time for one oscillation
The period of the sine function [tex]\(\sin(t)\)[/tex] is [tex]\(2\pi\)[/tex], but since we have [tex]\(\sin(3t)\)[/tex], the period is scaled by the factor inside the sine function. To find the period of [tex]\(3 \sin(t)\)[/tex], we use the formula [tex]\( \text{Period} = \frac{2\pi}{\text{frequency}} \)[/tex]. Here, the frequency is 3:
[tex]\[ \text{Period} = \frac{2\pi}{3} \][/tex]
Thus, the time required for one complete oscillation is:
[tex]\[ \boxed{2.094 \text{ seconds}} \][/tex]
