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Question
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
Solution
Let x and y denote the length and breadth in metres of the sheet of paper and A denote the area of the printed space.
Then, area of the sheet of paper = length × breadth
= xy
= 24
∴ y = `24/x` .......(i)
Also, length of the printed space = (x − 1) metres and its breadth = (y − 1.5) metres.
∴ Area of the printed space,
A = (x − 1)(y − 1.5)
= `(x - 1)(24/x - 1.5)` .......[From (i)]
= `24 - 1.5x - 24/x + 1.5`
= `25.5 - 1.5x - 24/x`
∴ `("dA")/("d"x) = 0 - 1.5 + 24/(x^2)`
= `-3/2 + 24/(x^2)`
∴ `"dA"/("d"x^2)` = 0 + 24(–2x –3)
= `-(48)/(x^3)`
Now, A is maximum, if `"dA"/("d"x)` = 0
∴ `-3/2 + 24/(x^2)` = 0
∴ `24/(x^2) = 3/2`
∴ x2 = `24 xx 2/3`
= 16
∴ x = 4 .......[∵ x > 0]
For x = 4,
`(("d"^2"A")/("d"x^2))_((x = 4)) = - 48/(x^3)`
= `-48/(4^3)`
= `-3/4 < 0`
Thus, A is maximum when x = 4.
From (i), we get
y = `24/x`
= `24/4`
= 6
Thus, the area of printed space is maximum when length and breadth of the sheet are 4 metres and 6 metres respectively.
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