MHT CET · Maths · Application of Derivatives
A firm is manufacturing 2000 items. It is estimated that the rate of change of production \(P\) with respect to additional number of workers \(x\) is given by \(\frac{d p}{d x}=100-12 \sqrt{x}\). If the firm employs 25 more workers, then the new level of production of items is
- A \(2500\)
- B \(3000\)
- C \(3500\)
- D \(4500\)
Answer & Solution
Correct Answer
(C) \(3500\)
Step-by-step Solution
Detailed explanation
\(\frac{d p}{d x}=100-12 \sqrt{x} \Rightarrow \int d p=\int(100-12 \sqrt{x}) d x\)
\(\Rightarrow p=100 x-12 \times \frac{2}{3} x^{3 / 2}+C\)
\(\Rightarrow p=100 x-8 x^{3 / 2}+200[\because \text { at } x=0, p=2000]\)
\(\Rightarrow p=100 \times 25-8 \times(25)^{3 / 2}+2000[\text { putting }\) \(x=25]\)
\(\Rightarrow p=2500-1000+2000=3500\)
\(\Rightarrow p=100 x-12 \times \frac{2}{3} x^{3 / 2}+C\)
\(\Rightarrow p=100 x-8 x^{3 / 2}+200[\because \text { at } x=0, p=2000]\)
\(\Rightarrow p=100 \times 25-8 \times(25)^{3 / 2}+2000[\text { putting }\) \(x=25]\)
\(\Rightarrow p=2500-1000+2000=3500\)
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