At present, a firm is manufacturing 2000 items.

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 $${{dp} \over {dx}} = 100 - 12\sqrt x .$$ If the firm employs $25$ more workers, then the new level of production of items is

Accepted Answer

Given, Rate of change is $${{dp} \over {dx}} = 100 - 12\sqrt x $$

$$ \Rightarrow dP = \left( {100 - 12\sqrt x } \right)dx$$

By intergrating $$\int {dP = \int {\left( {100 - 12\sqrt x } \right)} } dx$$

$$\int {dP}  = \int {\left( {100 - 12\sqrt x } \right)} dx$$

$P = 100x - 8{x^{3/2}} + C$

Given, when $x=0$ then $P=2000$

$$ \Rightarrow C = 2000$$

Now when $x$ $=25$then$P = 100 \times 25 - 8 \times {\left( {25} \right)^{3/2}} + 2000$$\,\,\,\,\,\,\,\,\,\,$$=4500-1000$$ \Rightarrow P = 3500$

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