The dimensions that maximize the area are, 96cm and 84cm.
What is area?
The measurement that expresses the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
Let the height of poster be h and breadth be b,
bh = 15630cm square........(1)
Height of the printed area=h-10-10 = h - 20
(decrement of 10 from top and bottom)
the breadth of the printed area b-6-6 = b-12
(decrement of 9 from both sides)
for maximum printed area:
A=(h-20)(b-12) should be maximum
[tex]A = (h - 20)(\frac{15360 }{h}-12)[/tex]
From equation (1)
[tex]A = 15360-12h-\frac{307200}{h} +240[/tex]
A = 15600 - 12h - (307200/h)
differentiate with respect to h (it should be=0)
[tex]\frac{dA}{dh}= 0-12 + \frac{307200}{h^2}[/tex]
[tex]12 = \frac{307200}{h} \\h = 160 cm[/tex]
from equation (1),
[tex]bh = 15630cm^2\\b = 96cm[/tex]
dimension of printed area,
= h - 20
= 160 - 20
= 140 cm
= b - 12
= 96 - 12
= 84 cm
Therefore, the dimensions that maximize the area are, 96cm and 84cm.
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