Answer:
23.92 cm wide by 35.87 cm high
Step-by-step explanation:
You want the dimensions of a poster of minimum area, given the printed portion has area 380 cm² and the margins are 4 cm on each side and 6 cm at top and bottom.
Setup
Let x represent the width of the printed area. Then x+8 is the width of the poster. The height of the printed area will be 380/x, and the height of the poster will be 380/x +12.
The area we want to minimize is the product of the width and height of the poster:
A = (x +8)(380/x +12)
Minimize
We can minimize the area by setting its derivative with respect to x equal to zero.
A' = (1)(380/x +12) +(x +8)(-380/x²)
A' = 12 -3040/x² = 0
x² = 3040/12 = 760/3
x = (2/3)√570 ≈ 15.9164
Now, we can find the desired dimensions:
x +8 ≈ 23.92
380/x +12 ≈ 35.87
The dimensions of the poster with minimum area are 23.92 cm wide by 35.87 cm high.
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Additional comment
If we don't combine factors so quickly, we can see that x = √(2/3·380) and the corresponding y-dimension is √(3/2·380). That is, the poster is 1.5 times as high as it is wide, and that ratio is equal to the ratio of margin widths.
Having seen this solution, you can work any similar problem without using any calculus.
