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# The Total Cost of Producing X Radio Sets per Day is Rs ( X 2 4 + 35 X + 25 ) and the Price per Set at Which They May Be Sold is Rs. ( 50 − X 2 ) . Ind the - CBSE (Science) Class 12 - Mathematics

#### Question

The total cost of producing x radio sets per  day is Rs $\left( \frac{x^2}{4} + 35x + 25 \right)$ and the price per set  at which they may be sold is Rs. $\left( 50 - \frac{x}{2} \right) .$ Find the daily output to maximum the total profit.

#### Solution

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

$\Rightarrow P = x\left( 50 - \frac{x}{2} \right) - \left( \frac{x^2}{4} + 35x + 25 \right)$

$\Rightarrow P = 50x - \frac{x^2}{2} - \frac{x^2}{4} - 35x - 25$

$\Rightarrow \frac{dP}{dx} = 50 - x - \frac{x}{2} - 35$

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

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

$\Rightarrow 15 - \frac{3x}{2} = 0$

$\Rightarrow 15 = \frac{3x}{2}$

$\Rightarrow x = \frac{30}{3}$

$\Rightarrow x = 10$

$\text { Now,}$

$\frac{d^2 P}{d x^2} = \frac{- 3}{2} < 0$

$\text{ So, profit is maximum if daily output is 10 items.}$

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Solution The Total Cost of Producing X Radio Sets per Day is Rs ( X 2 4 + 35 X + 25 ) and the Price per Set at Which They May Be Sold is Rs. ( 50 − X 2 ) . Ind the Concept: Graph of Maxima and Minima.
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