Let the paper size be "y" inches in length and "x" inches in width.
The length of the printed space would be "(y-2\\cdot3)" inches and width would be "(x-2\\cdot2)" inches.
Johnny is designing a rectangular poster to contain "24\\ in^2" of printing
Solve for "y"
Since the area of the paper of size "x" inches by "y" inches is "xy," let it be denoted as A
Then
Find the first derivative with respect to "x"
Find the critical number(s)
"4(x-4)-4x+(x-4)^2=0"
"4x-16-4x+x^2-8x+16=0"
Critical numbers: "x=0, x=8"
First derivative test
"If \\ x<0, A'(x)>0, A(x)\\ increases."
"If \\ 0<x<8, A'(x)<0, A(x)\\ decreases."
"If \\ x>8, A'(x)>0, A(x)\\ increases."
The function "A(x)" has the local maximum at "x=0."
The function "A(x)" has the local minimum at "x=8."
We consider "x>4." Hence the function "A(x)" has the absolute minimum for "x>4" at "x=8."
Find the length
We need to use the paper "8\\ in\\times 12 \\ in."
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