Let's break down each part of the problem and solve it step-by-step.
### Part (a) Find an equation [tex]\( v(t) \)[/tex] for the object velocity.
We are given the acceleration [tex]\( a(t) = 7 \sin(t) \)[/tex]. The acceleration is the derivative of velocity, i.e., [tex]\( v'(t) = a(t) \)[/tex].
To find the velocity [tex]\( v(t) \)[/tex], we need to integrate the acceleration function.
[tex]\[
v(t) = \int 7 \sin(t) \, dt
\][/tex]
The integral of [tex]\( 7 \sin(t) \)[/tex] is:
[tex]\[
v(t) = -7 \cos(t) + C
\][/tex]
To determine the constant [tex]\( C \)[/tex], we use the initial condition [tex]\( v(0) = -13 \)[/tex]:
[tex]\[
v(0) = -7 \cos(0) + C = -13
\][/tex]
Since [tex]\( \cos(0) = 1 \)[/tex], we can solve for [tex]\( C \)[/tex]:
[tex]\[
-7 \cdot 1 + C = -13
\][/tex]
[tex]\[
-7 + C = -13
\][/tex]
[tex]\[
C = -6
\][/tex]
Thus, the velocity function is:
[tex]\[
v(t) = -7 \cos(t) - 6
\][/tex]
### Part (b) Find the object's displacement from time 0 to time 3.
Displacement is the integral of velocity over the given time interval [0, 3].
[tex]\[
\text{Displacement} = \int_{0}^{3} v(t) \, dt
\][/tex]
Using the velocity function [tex]\( v(t) = -7 \cos(t) - 6 \)[/tex]:
[tex]\[
\text{Displacement} = \int_{0}^{3} (-7 \cos(t) - 6) \, dt
\][/tex]
After performing this integration, we obtain:
[tex]\[
\text{Displacement} = -18.9878 \, \text{meters}
\][/tex]
### Part (c) Find the total distance traveled by the object from time 0 to time 3.
The total distance traveled is the integral of the absolute value of the velocity function over the given time interval [0, 3].
[tex]\[
\text{Total Distance} = \int_{0}^{3} |v(t)| \, dt
\][/tex]
Using the velocity function [tex]\( v(t) = -7 \cos(t) - 6 \)[/tex], we compute:
[tex]\[
\text{Total Distance} = \int_{0}^{3} | -7 \cos(t) - 6 | \, dt
\][/tex]
After performing this integration, we obtain:
[tex]\[
\text{Total Distance} = 19.4292 \, \text{meters}
\][/tex]
### Summary of Results
(b) The object's displacement from time 0 to time 3 is:
[tex]\[
-18.9878 \, \text{meters}
\][/tex]
(c) The total distance traveled by the object from time 0 to time 3 is:
[tex]\[
19.4292 \, \text{meters}
\][/tex]
