Step 1 :The given equation is a differential equation. To find \(E(t)\), we need to integrate the given equation with respect to \(t\).
Step 2 :Integrating \(\frac{d E}{d t}=30-8 t\) with respect to \(t\), we get \(E(t) = -4t^2 + 30t + C\), where \(C\) is the constant of integration.
Step 3 :We can use the initial condition \(E(2) = 77\) to find the constant of integration.
Step 4 :Substituting \(t = 2\) and \(E(2) = 77\) into the equation, we get \(77 = -4(2)^2 + 30(2) + C\). Solving for \(C\), we find that \(C = 33\).
Step 5 :Substituting \(C = 33\) back into the equation, we get \(E(t) = -4t^2 + 30t + 33\).
Step 6 :\(\boxed{E(t) = -4t^2 + 30t + 33}\) is the operator's efficiency as a function of time.