Respuesta :
Answer:
Dimensions of the poster
Width  29.10 cm
Height 38.80 cm
A(min) = 1129.08 cm²
Step-by-step explanation:
Printed area = 390 cm² = Ap
Lets call x and y dimensions of printed area
x  width
y  height
Then Ap = x*y   and
y = Ap/x   ⇒   y = 390/x
Then  total area of the poster is:
A(t)  = ( x  + 12 ) * ( y + 16 )     and  y = 390/x
A as a function of x
A(x) = ( x  + 12 ) * ( 390/x + 16 )  ⇒  A(x) = 390 + 16x + 4680/x + 192
A(x) = 582 + 16x + 4680/x   (1)
Taking derivatives
A´(x) =  16 - (4680/x²)     ⇒ A´(x)  =  0
[ 16x² -4680] /x² = 0   16x²- 4680 = 0   ⇒   x² = 4680/16
x = 17.10 cm     and   y = 390/x    y = 390/17.10    y = 22.80 cm
A(t) = ( 17.10 + 12 ) * ( 22.80 + 16 )
A(t) = 29.10 * 38.80
A(t) = 1129.08 cm²
If we substitute  in equation (1) the value of x  ( 17.10) we see A(x) > 0
Then there is a minimun at the point x = 17.10
