Question 999498


x = width of the banner  
 and y = length  
 therefore xy = 27 

 Area of  centre, 
  
A = (x - 1)(y - 16/12) 
 = (x - 1)(27/x - 4/3) 
 = 27 - (4/3)x - 27/x + 4/3 
 = 85/3 - (4/3)x - 27/x 

 For f(x)=85/3 - (4/3)x - 27/x , to be maximum,

 f '(x) = 0 and f "(x) < 0 
 f '(x) = 0 
 = -4/3 + 27/x^2 = 0 
 = x^2 = 81/4 
 = x = 9/2 


 f "(x) = - 54/x^2 < 0 
 x = 9/4 => y = 27 * 4/9 = 12 

 length and width of banner = 12 ft. x 2 1/4 ft. 

12*2 1/4 = 27