Let's go through each part of the problem step by step.
### Part (a) - Find the velocity at time [tex]\( t \)[/tex]
The position of the particle is given by [tex]\( s = f(t) = 3t^3 + 2t + 9 \)[/tex].
The velocity [tex]\( v(t) \)[/tex] is the first derivative of the position function [tex]\( s(t) \)[/tex] with respect to time [tex]\( t \)[/tex]. Taking the derivative:
[tex]\[ v(t) = \frac{d}{dt}(3t^3 + 2t + 9) \][/tex]
Using the power rule for differentiation:
[tex]\[ \frac{d}{dt}(3t^3) = 9t^2 \][/tex]
[tex]\[ \frac{d}{dt}(2t) = 2 \][/tex]
[tex]\[ \frac{d}{dt}(9) = 0 \][/tex]
So, combining these results:
[tex]\[ v(t) = 9t^2 + 2 \][/tex]
Thus,
[tex]\[ v(t) = 9t^2 + 2 \frac{m}{s} \][/tex]
### Part (b) - Find the velocity at time [tex]\( t = 3 \)[/tex] seconds
To find the velocity at [tex]\( t = 3 \)[/tex] seconds, we substitute [tex]\( t = 3 \)[/tex] into the velocity function [tex]\( v(t) \)[/tex]:
[tex]\[ v(3) = 9(3)^2 + 2 \][/tex]
Calculating the value:
[tex]\[ = 9 \times 9 + 2 \][/tex]
[tex]\[ = 81 + 2 \][/tex]
[tex]\[ = 83 \][/tex]
Thus, the velocity at [tex]\( t = 3 \)[/tex] seconds is:
[tex]\[ 83 \frac{m}{s} \][/tex]
### Part (c) - Find the acceleration at time [tex]\( t \)[/tex]
The acceleration [tex]\( a(t) \)[/tex] is the first derivative of the velocity function [tex]\( v(t) \)[/tex] with respect to time [tex]\( t \)[/tex]. Taking the derivative of [tex]\( v(t) = 9t^2 + 2 \)[/tex]:
[tex]\[ a(t) = \frac{d}{dt}(9t^2 + 2) \][/tex]
Using the power rule for differentiation:
[tex]\[ \frac{d}{dt}(9t^2) = 18t \][/tex]
[tex]\[ \frac{d}{dt}(2) = 0 \][/tex]
So, combining these results:
[tex]\[ a(t) = 18t \][/tex]
Thus,
[tex]\[ a(t) = 18t \frac{m}{s^2} \][/tex]
### Part (d) - Find the acceleration at time [tex]\( t = 3 \)[/tex] seconds
To find the acceleration at [tex]\( t = 3 \)[/tex] seconds, we substitute [tex]\( t = 3 \)[/tex] into the acceleration function [tex]\( a(t) \)[/tex]:
[tex]\[ a(3) = 18(3) \][/tex]
Calculating the value:
[tex]\[ = 54 \][/tex]
Thus, the acceleration at [tex]\( t = 3 \)[/tex] seconds is:
[tex]\[ 54 \frac{m}{s^2} \][/tex]
So, summarizing the answers:
- Part (a) [tex]\( v(t) = 9t^2 + 2 \frac{m}{s} \)[/tex]
- Part (b) [tex]\( 83 \frac{m}{s} \)[/tex]
- Part (c) [tex]\( a(t) = 18t \frac{m}{s^2} \)[/tex]
- Part (d) [tex]\( 54 \frac{m}{s^2} \)[/tex]
