SOLUTION: A rectangular poster needs to have 2700 square centimeters of printed material surrounded by margins. The margins on the left and right edges need to be 4 cm thick, and the margins

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Question 1055475: A rectangular poster needs to have 2700 square centimeters of printed material surrounded by margins. The margins on the left and right edges need to be 4 cm thick, and the margins on the top and bottom edges need to be 3 cm thick. What should the dimensions of the overall poster be in order to minimize the total area?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
let +A+ = the total area of printed material
plus margins in cm2
Let +x+ = one of the dimensions of printed material
Let +y+ = the other dimension of printed material
(1) +A+=+%28+x+%2B+2%2A3+%29%2A%28+y+%2B+2%2A4+%29+ cm2
------------------------------------
(1) +A+=+%28+x+%2B+6+%29%2A%28+y+%2B+8+%29+ cm2
(2) +x%2Ay+=+2700+ cm2
------------------------------------
(2) +y+=+2700%2Fx+
Plug (2) into (1)
(1) +A+=+%28+x+%2B+6+%29%2A%28+2700%2Fx+%2B+8+%29+
(1) +A+=+2700+%2B+16200%2Fx+%2B+8x+%2B+48+
(1) +A+=+16200%2Fx+%2B+8x+%2B+2748+
-------------------------------------
Is this a calculus problem? That would be the only
way I know how to solve this.
Set the derivative of +A+ with respect to +x+ = +0+
+-16200%2Fx%5E2+%2B+8+=+0+
+16200%2Fx%5E2+=+8+
+x%5E2+=+16200%2F8+
+x%5E2+=+2025+
+x+=+45+
----------------
(2) +y+=+2700%2Fx+
(2) +y+=+2700%2F45+
(2) +y+=+60+
----------------------
+x+%2B+6+=+45+%2B+6+
+x+%2B+6+=+51+
and
+y+%2B+8+=+60+%2B+8+
+y+%2B+8+=+68+
----------------------------------
The dimensions of the overall
poster to minimize area is 51 x 68
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Here's the plot of +A%28x%29+:
+graph%28+400%2C+400%2C+-30%2C+100%2C+-500%2C+5000%2C+16200%2Fx+%2B+8x+%2B+2748+%29+
This looks OK -
+A%28+51+%29+=+51%2A68+
+A%28+51+%29+=+3468+ looks like a minimum