A poster is to have an area of 630 cm2 with 2.5 cm margins at the bottom and sides and a 5 cm margin at the top. Find the exact dimensions (in cm) that will give the largest printed area.
width : cm
height: cm
1
Expert's answer
2022-05-24T09:54:11-0400
Let x= width of the poster, let y= height of the poster.
Then
xy=630
The largest printed area will be
A=(x−2.5−2.5)(y−2.5−5)
Substitute
A=A(x)=(x−5)(x630−7.5)
=630−7.5x−x3150+37.5
=667.5−7.5x−x3150,5≤x≤84
Differentiate wih respect to x
A′=−7.5+x23150
Find the critical number(s)
A′=0=>−7.5+x23150=0
x=±420
Since 5≤x≤84, we take x=420
If 5≤x<420,A′>0,A increases.
If 420<x≤420,A′<0,A decreases.
The function A has the absolute maximum for 5≤x≤84 at x=420.
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