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

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length of page =16cm and width of page = 20 cm.

length of page =18cm and width of page = 30 cm.

length of page =15cm and width of page = 10 cm.

length of page =10cm and width of page = 10 cm.

answer is C.

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Detailed Solution

Given, area of page =

150 {cm}^{2}

Suppose, length of total page = x
Width of total page = y
Thus, the area,
A= length

\times

width,
150

= xy

\Rightarrow y = \frac{150}{x}

…..(1)
Now, length of printed matter =x−3 (margin at the top and bottom is 3 cm)
Width of printed matter =y−2 (margin at side is 2 cm)
Therefore, area of printed matter,

A_{p} = (x - 3) \times (y - 2)

\begin{matrix} A_{p} = 150 - 3 (\frac{150}{x}) - 2 x + 6 \\ A_{p} = 156 - (\frac{450}{x}) - 2 x \end{matrix}

Differentiate area with respect to x and equate the function to zero as,

\begin{matrix} \frac{d A_{p}}{dx} = \frac{450}{x^{2}} - 2 = 0 \\ \Rightarrow x^{2} = 225 \end{matrix}

x = 15 cm

Put this value in equation 1,

y = \frac{150}{15}

y = 10 cm)

Again, differentiate the area with respect to x,

\frac{d^{2} A_{p}}{{dx}^{2}} = - \frac{900}{x^{3}} < 0

Hence area is maximum.
Therefore, length of page =15 cm and width of page = 10 cm.
Correct option is 3.

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