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
then the discontinuity point is at 
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