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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 dP/dx = 100 − 12√x. If the firm employs 25 more workers, then the new level of production of items is
  • a)
    3500
  • b)
    4500
  • c)
    2500
  • d)
    3000
Correct answer is option 'A'. Can you explain this answer?
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At present, a firm is manufacturing 2000 items. It is estimated that ...
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At present, a firm is manufacturing 2000 items. It is estimated that ...
Given Information:
- The rate of change of production P with respect to additional number of workers x is given by dP/dx = 100 − 12√x.
- The firm is currently manufacturing 2000 items.

To Find:
- The new level of production of items if the firm employs 25 more workers.

Step-by-Step Solution:

Step 1: Find the antiderivative of dP/dx to get P.
- Integrate both sides of the given equation with respect to x:
∫dP/dx dx = ∫(100 − 12√x) dx
- The integral of dP/dx with respect to x gives us P:
P = 100x - 12 * (2/3) * x^(3/2) + C, where C is the constant of integration.

Step 2: Use the given information to find the constant of integration.
- We know that the firm is currently manufacturing 2000 items, so we can substitute this value into the equation:
2000 = 100(2000) - 12 * (2/3) * (2000)^(3/2) + C
- Simplify and solve for C:
2000 = 200000 - 12 * (2/3) * 2000^(3/2) + C
C = 2000 - 200000 + 12 * (2/3) * 2000^(3/2)
C = 2000 - 200000 + 16000
C = -181000

Step 3: Substitute the value of C into the equation for P.
P = 100x - 12 * (2/3) * x^(3/2) - 181000

Step 4: Find the new level of production when 25 more workers are employed.
- Substitute x = 2000 + 25 into the equation for P:
P = 100(2025) - 12 * (2/3) * (2025)^(3/2) - 181000
P = 202500 - 12 * (2/3) * 2025^(3/2) - 181000
P = 202500 - 12 * (2/3) * (45)^3/2 - 181000
P = 202500 - 12 * (2/3) * 45 * 45 - 181000
P = 202500 - 12 * 30 * 45 - 181000
P = 202500 - 16200 - 181000
P = 202500 - 197200
P = -106700

Step 5: Interpret the result.
- The negative value for P indicates that the new level of production is not possible or meaningful in this context.
- Therefore, we can conclude that there may be a mistake in the given information or calculations.

Final Answer:
The new level of production of items when the firm employs 25 more workers cannot be determined with the given information.
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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 dP/dx = 100 − 12√x. If the firm employs 25 more workers, then the new level of production of items isa)3500b)4500c)2500d)3000Correct answer is option 'A'. Can you explain this answer?
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