Final answer:
The current through the inductor can be found by differentiating the given voltage equation and integrating the result. The equation for the current through the inductor is i(t) = 3t² + 2t + 9 A.
Step-by-step explanation:
To determine the current through the inductor, we can differentiate the given voltage equation with respect to time. Given v(t) = (3t² + 2t + 4) V, we can find di(t)/dt by taking the derivative of v(t). So, di(t)/dt = d/dt(3t² + 2t + 4) = 6t + 2 A/s. Now, we can integrate di(t)/dt to find the current i(t). Given i(0) = 9 A, we can integrate di(t)/dt = 6t + 2 over the interval from 0 to t to find i(t). Integrating 6t + 2 with respect to t gives us i(t) = 3t² + 2t + C, where C is a constant of integration. To solve for C, we can use the initial condition i(0) = 9 A. Substituting t = 0 and i(t) = 9 into the equation i(t) = 3t² + 2t + C, we get 9 = 0 + 0 + C, so C = 9. Therefore, the current i(t) through the inductor is i(t) = 3t² + 2t + 9 A.