A particle is moving with acceleration $a(t)=30 t+12$. its position at time $t=0$ is $s(0)=17$ and its velocity at time $t=0$ is $v(0)=12$. What is its position at time $t=7$ ?

Step 1 ：The acceleration of the particle is given by the function \(a(t) = 30t + 12\).

Step 2 ：We can find the velocity of the particle as a function of time by integrating the acceleration function. This gives us \(v(t) = 15t^2 + 12t + C\), where \(C\) is the constant of integration.

Step 3 ：Given that the velocity of the particle at time \(t=0\) is \(v(0) = 12\), we can solve for \(C\) in the velocity function. This gives us \(C = 12\).

Step 4 ：Therefore, the velocity of the particle as a function of time is \(v(t) = 15t^2 + 12t + 12\).

Step 5 ：We can find the position of the particle as a function of time by integrating the velocity function. This gives us \(s(t) = 5t^3 + 6t^2 + 12t + D\), where \(D\) is the constant of integration.

Step 6 ：Given that the position of the particle at time \(t=0\) is \(s(0) = 17\), we can solve for \(D\) in the position function. This gives us \(D = 17\).

Step 7 ：Therefore, the position of the particle as a function of time is \(s(t) = 5t^3 + 6t^2 + 12t + 17\).

Step 8 ：Substituting \(t = 7\) into the position function gives us the position of the particle at time \(t = 7\). This gives us \(s(7) = 2110\).

Step 9 ：Final Answer: The position of the particle at time \(t=7\) is \(\boxed{2110}\).