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Manufacturer Can Sell X Items at a Price of Rupees ( 5 − X 100 ) Each. the Cost Price is Rs ( X 5 + 500 ) . Find the Number of Items He Should Sell to Earn Maximum Profit. - CBSE (Arts) Class 12 - Mathematics

Question

Manufacturer can sell x items at a price of rupees $\left( 5 - \frac{x}{100} \right)$ each. The cost price is Rs  $\left( \frac{x}{5} + 500 \right) .$ Find the number of items he should sell to earn maximum profit.

Solution

$\text { Profit =S.P. - C.P.}$

$\Rightarrow P = x\left( 5 - \frac{x}{100} \right) - \left( 500 + \frac{x}{5} \right)$

$\Rightarrow P = 5x - \frac{x^2}{100} - 500 - \frac{x}{5}$

$\Rightarrow \frac{dP}{dx} = 5 - \frac{x}{50} - \frac{1}{5}$

$\text { For maximum or minimum values of P, we must have }$

$\frac{dP}{dx} = 0$

$\Rightarrow 5 - \frac{x}{50} - \frac{1}{5} = 0$

$\Rightarrow \frac{24}{5} = \frac{x}{50}$

$\Rightarrow x = \frac{24 \times 50}{5}$

$\Rightarrow x = 240$

$\text { Now, }$

$\frac{d^2 P}{d x^2} = \frac{- 1}{50} < 0$

$\text { So, the profit is maximum if 240 items are sold.}$

Notes

The solution given in the book is incorrect. The solution here is created according to the question given in the book.

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Solution Manufacturer Can Sell X Items at a Price of Rupees ( 5 − X 100 ) Each. the Cost Price is Rs ( X 5 + 500 ) . Find the Number of Items He Should Sell to Earn Maximum Profit. Concept: Graph of Maxima and Minima.
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