Differential Equations
Differential Equation
nta_abhyas_2025
Grade None
Question:
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
Step-by-Step Solution
Key Concept: Integration of a differential equation with initial conditions determines the particular solution.
Integrating both sides: $P = \int (100 - 12\sqrt{x})dx = 100x - 8x^{3/2} + C$. Using the initial condition $P = 2000$ when $x = 0$, we get $C = 2000$. Therefore, $P = 100x - 8x^{3/2} + 2000$. When $x = 25$: $P = 100(25) - 8(25)^{3/2} + 2000 = 2500 - 8(125) + 2000 = 2500 - 1000 + 2000 = 3500$.
Correct Answer: 3500