Question 11 Mark
Assertion $(A)$ : If manufacturer can sell $x$ items at a price of $₹ \left(5-\frac{x}{100}\right)$ each.
The cost price of $x$ items is $₹ \left(\frac{x}{5}+500\right)$
Then, the number of items he should sell to earn maximum profit is $240$ items.
Reason $(R)$ : The profit for selling x items is given by $\frac{24}{5} x-\frac{x^2}{100}-300$
The cost price of $x$ items is $₹ \left(\frac{x}{5}+500\right)$
Then, the number of items he should sell to earn maximum profit is $240$ items.
Reason $(R)$ : The profit for selling x items is given by $\frac{24}{5} x-\frac{x^2}{100}-300$
Answer
View full question & answer→$(c) \ A$ is true but $R$ is false.
Explanation : Let $S\ (x)$ be the selling price of $x$ items and let $C\ (x)$ be the cost price of $x$ items.
Then, we have
$S(x)=\left(5-\frac{x}{100}\right) x=5 x-\frac{x^2}{100}$
and $C(x)=\frac{x}{5}+500$
Thus, the profit function $P(x)$ is given by
$P ( x )= S ( x )- C ( x )=5 x -\frac{x^2}{100}-\frac{x}{5}-500$
i.e. $ P ( x )=\frac{24}{5} x-\frac{x^2}{100}-500$
On differentiating both sides $\text{w.r.t. x,}$ we get
$P ^{\prime}( x )=\frac{24}{5}-\frac{x}{50}$
Now, $P ^{\prime}( x )=0$ gives $x =240$.
Also, $P ^{\prime}( x )=\frac{-1}{50}$.
So, $P ^{\prime}(240)=\frac{-1}{50}<0$
Thus $, x = 240$ is a point of maxima.
Hence, the manufacturer can earn maximum profit, if he sells $240$ items.
Explanation : Let $S\ (x)$ be the selling price of $x$ items and let $C\ (x)$ be the cost price of $x$ items.
Then, we have
$S(x)=\left(5-\frac{x}{100}\right) x=5 x-\frac{x^2}{100}$
and $C(x)=\frac{x}{5}+500$
Thus, the profit function $P(x)$ is given by
$P ( x )= S ( x )- C ( x )=5 x -\frac{x^2}{100}-\frac{x}{5}-500$
i.e. $ P ( x )=\frac{24}{5} x-\frac{x^2}{100}-500$
On differentiating both sides $\text{w.r.t. x,}$ we get
$P ^{\prime}( x )=\frac{24}{5}-\frac{x}{50}$
Now, $P ^{\prime}( x )=0$ gives $x =240$.
Also, $P ^{\prime}( x )=\frac{-1}{50}$.
So, $P ^{\prime}(240)=\frac{-1}{50}<0$
Thus $, x = 240$ is a point of maxima.
Hence, the manufacturer can earn maximum profit, if he sells $240$ items.