Answer:
Dimensions of the poster are:
w = 18 in
Y = 36 in
Step-by-step explanation:
Print area of the rectangular poster is 512 in²
Let call "x" and "y" dimensions for the print area of the poster then
A(p) = Print area of the poster = x*y
512 = x*y ⇒ y = 512/x
Total area of the poster is:
A(t) = ( y + 4 ) * ( x + 2 )
A(t) = y*x +2*y +4*x + 8 And as y = 512/x
Total area of the poster as a function of x is:
A(x) =( 512/x)*x + 2* (512/x) + 4*x + 8
A(x) = 512 + 1024/x + 4*x + 8 ⇒ A(x) = 520 + 1024/x + 4*x
Taking derivatives on both sides of the equation we get:
A´(x) = - 1024/x² + 4
A´(x) = 0 ⇒ - 1024 /x² = -4 ⇒ 4*x² = 1024
x² = 1024/4 ⇒ x² = 256
x = 16 inches
And y = 512/x ⇒ y = 512/16 ⇒ y = 32 inches
So we found x and y dimensions of the print area, then the dimensions of the poster are:
w = x + 2 ⇒ w = 16 + 2 w = 18 in
Y = y + 4 ⇒ Y = 32 + 4 Y = 36 in
