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(C) Given the rate of change of production: $\frac{dP}{dx} = 100 - 12\sqrt{x}$.Integrating both sides with respect to $x$: $\int dP = \int (100 - 12x^

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$3500$

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(C) Given the rate of change of production: $\frac{dP}{dx} = 100 - 12\sqrt{x}$.Integrating both sides with respect to $x$: $\int dP = \int (100 - 12x^{1/2}) dx$.$P = 100x - 12 	imes \frac{x^{3/2}}{3/2} + C = 100x - 8x^{3/2} + C$.Initially,when $x = 0$,the production $P = 2000$. Substituting these values: $2000 = 100(0) - 8(0)^{3/2} + C$,which gives $C = 2000$.So,the production function is $P(x) = 100x - 8x^{3/2} + 2000$.For $x = 25$ additional workers,the new production level is: $P(25) = 100(25) - 8(25)^{3/2} + 2000$.$P(25) = 2500 - 8(125) + 2000$.$P(25) = 2500 - 1000 + 2000 = 3500$.

$A$ firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ with respect to the 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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