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
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