At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $$P$$ w.r.t. additional number of workers $$x$$ is given by $$\frac{{dP}}{{dx}} = 100 - 12\sqrt x .$$     If the firm employs 25 more workers, then the new level of production of items is-

Solution :
Given, Rate of change is
$$\eqalign{ & \frac{{dP}}{{dx}} = 100 - 12\sqrt x \cr & \Rightarrow dP = \left( {100 - 12\sqrt x } \right)dx \cr} $$
By integrating
$$\eqalign{ & \Rightarrow \int {dP = \int {\left( {100 - 12\sqrt x } \right)dx} } \cr & P = 100x - 8{x^{\frac{3}{2}}} + C \cr} $$
Given, when $$x=0$$   then $$P =2000$$
$$ \Rightarrow C = 2000$$
Now when $$x= 25$$  then
$$\eqalign{ & P = 100 \times 25 - 8 \times {\left( {25} \right)^{\frac{3}{2}}} + 2000 \cr & \Rightarrow P = 4500 - 1000 \cr & \Rightarrow P = 3500 \cr} $$