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
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
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Separate variables and integrate \(dP = (100 - 12\sqrt{x})\,dx\) with P = 2000 at x = 0, then put x = 25.
Updated On: Nov 13, 2026
- 2500
- 3000
- 3500
- 4500
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The Correct Option is C
Solution and Explanation
Given \(\frac{dP}{dx} = 100 - 12\sqrt{x}\), so \(dP = (100 - 12\sqrt{x})\,dx\).
Integrate: \(P = 100x - 8x^{3/2} + C\).
At present x = 0 and P = 2000, so C = 2000.
Hence \(P = 100x - 8x^{3/2} + 2000\).
For x = 25: \(P = 2500 - 8\cdot 125 + 2000 = 2500 - 1000 + 2000 = 3500\).
Answer: (3) 3500
Integrate: \(P = 100x - 8x^{3/2} + C\).
At present x = 0 and P = 2000, so C = 2000.
Hence \(P = 100x - 8x^{3/2} + 2000\).
For x = 25: \(P = 2500 - 8\cdot 125 + 2000 = 2500 - 1000 + 2000 = 3500\).
Answer: (3) 3500
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