An employee's monthly productivity $M$, in number of units produced, is found to be a function of the number $t$ of years of service. For a certain product, a productivity function is shown below. Find the maximum productivity and the year in which it is achieved. \[ M(t)=-4 t^{2}+192 t+200,0 \leq t \leq 40 \] The maximum productivity is achieved in year

Step 1 ：An employee's monthly productivity $M$, in number of units produced, is found to be a function of the number $t$ of years of service. For a certain product, a productivity function is shown below. Find the maximum productivity and the year in which it is achieved.

Step 2 ：The productivity function is given by \(M(t)=-4 t^{2}+192 t+200\), where $t$ is the number of years of service and $M$ is the monthly productivity.

Step 3 ：The maximum productivity is achieved when the derivative of the productivity function is zero. This is because the derivative of a function gives the rate of change of the function, and when this rate of change is zero, it means the function is at a maximum or minimum.

Step 4 ：In this case, since the function is a downward-opening parabola (because the coefficient of $t^2$ is negative), the maximum will be achieved when the derivative is zero.

Step 5 ：Let's find the derivative of the productivity function: \(M'(t) = 192 - 8t\).

Step 6 ：Setting the derivative equal to zero gives us the year in which maximum productivity is achieved: \(192 - 8t = 0\). Solving for $t$ gives $t = 24$.

Step 7 ：Final Answer: The maximum productivity is achieved in year \(\boxed{24}\).