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The Total Area of a Page is 150 Cm2. the Combined Width of the Margin at the Top and Bottom is 3 Cm and the Side 2 Cm. - CBSE (Arts) Class 12 - Mathematics

Question

The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?

Solution

$\text { Let x andybe the length and breadth of the rectangular page, respectively. Then,}$

$\text { Area of the page } = 150$

$\Rightarrow xy = 150$

$\Rightarrow y = \frac{150}{x} . . . \left( 1 \right)$

$\text { Area of the printed matter }=\left( x - 3 \right)\left( y - 2 \right)$

$\Rightarrow A = xy - 2x - 3y + 6$

$\Rightarrow A = 150 - 2x - \frac{450}{x} + 6$

$\Rightarrow \frac{dA}{dx} = - 2 + \frac{450}{x^2}$

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

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

$\Rightarrow - 2 + \frac{450}{x^2} = 0$

$\Rightarrow 2 x^2 = 450$

$\Rightarrow x = 15$

$\text { Substituting the value of x in }\left( 1 \right), \text { we get }$

$y = 10$

$\text { Now },$

$\frac{d^2 A}{d x^2} = \frac{- 900}{x^3}$

$\Rightarrow \frac{d^2 A}{d x^2} = \frac{- 900}{\left( 15 \right)^3}$

$\Rightarrow \frac{d^2 A}{d x^2} = \frac{- 900}{3375} < 0$

$\text { So, area of the printed matter is maximum when x = 15 and y = 10 } .$

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Solution The Total Area of a Page is 150 Cm2. the Combined Width of the Margin at the Top and Bottom is 3 Cm and the Side 2 Cm. Concept: Graph of Maxima and Minima.
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