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JEE Main 2013

JEE Main 2013 Mathematics question on Differential Equations

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production PP with respect to additional number of workers xx is given by dPdx=100−12x\frac{dP}{dx} = 100 - 12\sqrt{x}. If the firm employees 25 more workers, then the new level of production of items is
  1. A.
    2500
  2. B.
    3000
  3. C.
    3500
  4. D.
    4500
Show answer and solution

Correct option: C

Solution

Integrate dP/dx from x = 0 to x = 25:

ΔP = ∫₀²⁵ (100 - 12√x) dx = [100x - 12·(2/3)x^(3/2)]₀²⁵ = [100x - 8x^(3/2)]₀²⁵

= 100(25) - 8(25)^(3/2) = 2500 - 8(125) = 2500 - 1000 = 1500

New production = 2000 + 1500 = 3500.

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