Answer:
The short side is _15___ft and the long side is ___30___ ft.
Step-by-step explanation:
As fence is built in a rectangular area, so we can consider
Let
  x be the length of the rectangle  
 y be width of the rectangle
Given area of rectangle is = 450 ft²
Formula for area of rectangle = Length x width  
      450 ft² =  xy  
Solve for y
y = 450/x            
 now according to given condition  
three sides of the fence costs $5 per foot and for the fourth side costs$15 per foot.
 We have two condition either the fourth side be x or y
So condition 1:  Three sides =(x,y,x)       4-th side = y.
So  we can write as 5x,5y,5x  and  15 y  
Cost C = 5x +5y+5x+15y
          = 10x+ 5y+15y
          = 5(2x+y) +15y----------------equation 1
            = 10x +20y
Adding value of y 450/x
         = 10x + 20(450/x)
   = 10x + 9000/x
For minimum cost, we can consider the cost to be 0
  0 = 10x + 9000/x
Dividing and multiplying by -x/x
0 = -10 +9000/x²
10 = 9000/x²
  10x² = 9000/  
  x²= 900
x = 30
so y = 450/x = 450/30= 15 ft  
so adding the values of x and y in equation 1 we will have
cost C= 5(2x+y) +15y----------------equation 1
cost is          = 5(2(30)+15) +15(15)
                       =  $600 is the cost  
X= 30 y =15
So condition 2:  Three sides =(y,x,y)       4-th side = x.
So  we can write as 5y,5x,5y  and  15 x
Cost C = 5y +5x+5y+15x
          = 5x+ 10y+15x
          = 5(x+2y) +15x----------------equation 2
            = 20x +10y
Adding value of y= 450/x
         = 20x + 10(450/x)
   = 20x + 4500/x
For minimum cost, we can consider the cost to be 0
  0 = 20x + 4500/x
Dividing and multiplying by -x/x
0 = -20 +4500/x²
20 = 4500/x²
  20x² = 4500
  x²=  4500/20= 225
x = 15
so y = 450/x = 450/15= 30 ft  
so adding the values of x and y in equation 2 we will have
cost C= = 5(x+2y) +15x----------------equation 2
cost is          = 5(15+2(30) +15(15)
                       =  $600 is the cost  
y= 30 x=15
so from both conditions satisfy the cost and the two sides are known as length and width
so dimension will be 15 ft by 30 ft