The top and bottom margins of a poster are each 6 cm and the side margins are ea

Jason Farmer

Jason Farmer

Answered question

2021-10-09

The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm , find the dimensions of the poster with the smallest area.

Answer & Explanation

Obiajulu

Obiajulu

Skilled2021-10-10Added 98 answers

Step 1
Let x,y be the width and height respectively of the printed area, which has fixed area
xy=384cm2y=384x
The total height of the poster including 6 cm margins at the top and bottom is y+12
The total width of the poster including 4 cm margins at sides is x+8
The area of the total poster is
A=(x+8)(y+12)
Step 2
Substitute for y in the A equation, find A
A=(x+8)(384x+12)
=384+12x+3072x+96
=12x+3072x+480
A=123072x2
Step 3
Find where A=0
0=123072x2
3072x2=12
307212=x2
x2=256
x=16  (ignone negative root)
Since  A=2(3072)x3  is positive for  x>0,  A is concave up and  x=16  is a minimum
Step 4
Thecorrespondgyvalueisy=38416=24cm
The total poster width is
x+8=16+8=24cm
The total poster height is
y+12=24+12=36
Result
24 cm by 36 cm

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