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JEE MAIN Mathematics QUESTION #6003
Question 1

A manufacturing firm currently produces 2000 items. The rate of change of production $P$ with respect to additional workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm hires 25 additional workers, what will be the new level of production?

  • 2500
  • 3000
  • 3500✔️
  • 4500
Correct Answer Explanation

We integrate $\frac{dP}{dx} = 100 - 12\sqrt{x}$ from $x = 0$ to $x = 25$:

$P = \int_0^{25}(100 - 12\sqrt{x})\,dx = \left[100x - 12 \cdot \frac{2}{3}x^{3/2}\right]_0^{25}$

$= \left[100x - 8x^{3/2}\right]_0^{25} = 100(25) - 8(25)^{3/2} = 2500 - 8(125) = 2500 - 1000 = 1500$

New production $= 2000 + 1500 = \mathbf{3500}$