The total poster area is:
A = (l + 20) (w + 12)
where l and w is the length and width of the printed area
24000 = (l + 20) (w + 12)
l = [24000/(w + 12)] – 20
The printed area is simply:
a = l w
substituting l:
a = ([24000/(w + 12)] – 20) w
a = (24000w)(w + 12)^-1 – 20 w
Taking the first derivative da/dw:
da/dw = (24000)(w + 12)^-1 - (24000w)(w + 12)^-2 – 20
Set da/dw = 0:
(24000)/(w + 12) - (24000w)/(w + 12)^2 – 20 = 0
Multiply everything by (w + 12)^2:
24000 (w + 12) – 24000w - 20(w + 12)^2 = 0
24000w + 288000 – 24000w – 20 (w^2 + 24w + 144) = 0
-20 w^2 – 480 w + 285120 = 0
w^2 + 24 w = 14256
Completing the square:
(w + 12)^2 = 14256 + 12^2
w + 12 = ±120
w = 108 cm
l = [24000/(w + 12)] – 20
l = [24000/(108 + 12)] – 20
l = 180 cm
Therefore the dimensions of the whole poster are:
w + 12 = 120 cm
l + 20 = 200 cm
The poster should be 200 cm x 120 cm
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