Question

The total area of a page is 150 $c{m^2}$.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 ?

Answer 1

Hint: Here we will consider the total area of the page as $A=xy$ . Then we will consider the printed area as $A= (x-3)(y-2)$ after leaving margins using differentiation.

Complete step-by-step answer:

Let the length of the page be x and breadth be y i.e…,$xy = 150 \to (1)$.
After leaving the margins, the dimensions are x-3 and y-2, where Area A of printed matter is given by
$
  A = (x - 3)(y - 2) \\
  A = xy - 3y - 2x + 6 \\
$
Now using equation (i), we get
$A = 150 - \dfrac{{450}}{x} - 2x + 6$
Now, for Area to be maximum,
$
  \dfrac{{dA}}{{dx}} = 0 \\
   \Rightarrow \dfrac{{450}}{{{x^2}}} - 2 = 0 \\
   \Rightarrow x = \pm 15 \\
   \Rightarrow x = 15 \\
$
Now, $\dfrac{{{d^2}A}}{{d{x^2}}} < 0$
Hence, A is maximum at x=15
Now putting the value of x in equation (i), we get
$y = 10$
Note: To solve such questions, we should first use the formula of area, with an adequate knowledge of boundary conditions. Students make mistakes by considering the area as $A=(x-2)(y-3)$ . So one must take care while solving this type of problem.