Answer:
24cm by 36cm
Step-by-step explanation:
You may have written the values twice in the above question.
Let 'a' represent width of the poster and 'b' represent the height.
'A' represent the area of the poster to be minimized.
ab= 384cm²
b= 384/x
total height including top and bottom margins of a poster 6 cm each
b+12
total width including the side margins 4 cm:
a+8
Therefore, Area of total poster would be defined as
A= (b+12)(a+8)
Substituting 'b' from above in equation on Area, we have
A=(a+8)(384/x + 12)
A= 384+12a+ (3072/a) + 96
A= 12x + 3072/x + 480
A'= 12 - 3072/x² (consider it A' for now)
considering A' =0
0= 12 - 3072/x²
3072/x² = 12
x²= 3071/12
x²=256
x=+-16
we'll ignore the negative root
therefore, x-16
Since A''= 2(3072)/x³ will be positive for x>0, A is concave up and x=16 is a minimum.
The total value of corresponding b will be,
b= 384/16 = 24cm
and the total width of the poster will be
x+8= 16+8= > 24cm
and the total height will be y+12
24+12=> 36cm
thus, the dimensions of the poster of smallest area is 24cm by 36cm
