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?


Let ABCD be the total page and A’B’C’D’ be the area in which the matter is printed.


Let ‘l’ and ‘b’ be the length and breadth of the total page.


We know that area of the rectangle is l×b


From the problem,


⇒ lb = 150cm2 …… (1)


⇒ PQ + RS = 3cm


⇒ WX + YZ = 2cm


Let ‘l’’ and ‘b’’ be the length and breadth of the area in which the matter is printed,


From the figure,


⇒ l’ = (l - 2)cm


⇒ b’ = (b - 3)cm


⇒ Area of the printed matter (A) = (l - 2)(b - 3)cm2


From (1)


⇒


⇒


⇒


We need Area of the printed matter maximum and let us take A as the function of l


We know that for maxima and minima,


⇒


⇒


⇒


⇒ (∵ l2>0)


⇒


⇒ l = 10cm (since length is an positive quantity)


Differentiating A again,


⇒


⇒


At l = 10


⇒


⇒ <0(Maxima)


We get the area of the printed matter maximum for l = 10cm


Let’s find the corresponding breadth using eq(1)


⇒ 10b = 150


⇒ b = 15cm


∴ The dimensions required for the printed area to be maximum is l = 10cm and b = 15cm.


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