Let's go through the given problem step-by-step:
### Part (a): Finding the Total Displacement
The total displacement of a particle is found by integrating its velocity function over the given time interval.
Given the velocity function:
[tex]\[ v(t) = (t-5)(t-10)(t-14) \][/tex]
We need to integrate this function from [tex]\( t = 0 \)[/tex] to [tex]\( t = 14 \)[/tex]:
[tex]\[ D = \int_0^{14} v(t) \, dt \][/tex]
Performing this integration over the interval from [tex]\( 0 \leq t \leq 14 \)[/tex], we find the total displacement.
From the calculations, the total displacement is:
[tex]\[ -\frac{3724}{3} \text{ ft} \][/tex]
When we convert this to decimal form and round to two decimal places, we get:
[tex]\[ -1241.33 \text{ ft} \][/tex]
### Part (b): Finding the Total Distance Traveled
The total distance traveled by the particle is the integral of the absolute value of the velocity function over the given time interval. This accounts for any changes in direction by ensuring all contributions to the distance are positive.
To find the total distance, we integrate the absolute value of the velocity function:
[tex]\[ \text{Distance} = \int_0^{14} |v(t)| \, dt \][/tex]
From the calculations, integrating the absolute value over the interval from [tex]\( 0 \leq t \leq 14 \)[/tex], we find the total distance traveled.
The total distance found is:
[tex]\[ 1512.17 \text{ ft} \][/tex]
### Summary
To summarize the answers:
(a) The total displacement of the particle after 14 seconds is:
[tex]\[ -1241.33 \text{ ft} \][/tex]
(b) The total distance the particle traveled in the first 14 seconds is:
[tex]\[ 1512.17 \text{ ft} \][/tex]
