Answer:
[tex]s(8)=2186[/tex]
Step-by-step explanation:
A particle is moving with acceleration modeled by the function:
[tex]a(t)=18t+16[/tex]
We are given that its position s(t) at t = 0 is 10 and that its velocity v(t) at t = 0 is 16.
And we want to find its position at t = 8.
Velocity is the integral of the acceleration. Hence:
[tex]\displaystyle v(t)=\int a(t)\, dt=\int 18t+16\, dt[/tex]
Find the velocity. Remember the constant of integration!
[tex]v(t)=9t^2+16t+C[/tex]
Since v(t) is 16 when t = 0:
[tex](16)=9(0)^2+16(0)+C\Rightarrow C=16[/tex]
Hence, our velocity is given by:
[tex]v(t)=9t^2+16t+16[/tex]
Position is the integral of the velocity. Hence:
[tex]\displaystyle s(t)=\int v(t)\, dt=\int 9t^2+16t+16\, dt[/tex]
Integrate:
[tex]\displaystyle s(t)=3t^3+8t^2+16t+C[/tex]
s(t) is 10 when t = 10. Hence:
[tex]C=10[/tex]
So, our position function is:
[tex]s(t)=3t^3+8t^2+16t+10[/tex]
The position at t = 8 will be:
[tex]s(8)=3(8)^3+8(8)^2+16(8)+10=2186[/tex]
