. A poster must have 32 sq.in. of printed materials with margins of 4 inches each at the top and 2 inches at each side. Find the dimensions of the whole poster if its area is maximum.
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Expert's answer
2022-06-13T23:27:15-0400
Let x= the length of printed matter, y=the width of printed matter.
Given xy=32.
The length of a poster will be x+2(2), the width of a poster will be y+4.
The area of a poster is A=(x+4)(y+4).
xy=32=>y=x32
Substitute
A=A(x)=(x+4)(x32+4),x>0A(x)=32+4x+x128+16
Find the derivative with respect to x
A′(x)=(48+4x+x128)′=4−x2128
Find the critical number(s)
A′(x)=0=>4−x2128=0x2=32x1=−42,x2=42
If 0<x<42,A′(x)>0,A(x) increases.
If x>42,A′(x)<0,A(x) decreases.
The function A(x) has a local maximum at x=42.
Since the function A(x) has the only extremum for x>0, then the function A(x) has the absolute maximum at x=42 for x>0.
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