question_answer

Profit fnction of a company is given as \[P(x)=\frac{24x}{5}-\frac{{{x}^{2}}}{100}-500,\] where x is the number of units produced. What is the maximum profit of the company? Company feels its social responsibility and decided to contribute 10% of his profit for the orphanage. What is the amount contributed by the company for the charity? Justify that every company should do it.

Answer:

Given, profit function, \[P(x)=\frac{24x}{5}-\frac{{{x}^{2}}}{100}-500\] ?(i) On differentiating Eq. (i) w.r.t. x, we get             \[P'(x)=\frac{24}{5}-\frac{2x}{100}\] For maximum or minimum put P?(x) = 0 \[\Rightarrow \]   \[\frac{24}{5}-\frac{2x}{100}=0\] \[\Rightarrow \] \[\frac{24}{5}=\frac{2x}{100}\] \[\Rightarrow \]   \[x=\frac{24\times 100}{5\times 2}\] \[\Rightarrow \] x = 240 Now, differentiating Eq. (ii) w.r.t. x, we get             \[P''(x)=-\frac{2}{100}\] \[\Rightarrow \] \[P''(240)=-\frac{2}{100}<0\]Hence,. P(x) is maximum when x = 240.         Now, maximum profit \[=\frac{24}{5}\times 240-\frac{{{(240)}^{2}}}{100}-500\] \[=\frac{5760}{5}-\frac{57600}{100}-500\] \[=\frac{5760}{5}-576-500=\frac{5760}{5}-1076\]             \[=\frac{5760-5380}{5}=\frac{380}{5}=Rs.\,\,76\] Now, the amount contributed by the company for the charity \[=Rs.\,\,\left( 76\times \frac{10}{100} \right)=Rs.\,\,7.60\] Value Yes, every company should do it, as it is good for society.