Answer:
x = 20√2 ft and y = 40/√2 ft
Step-by-step explanation:
Let; x = length of steel fencing
y = length of a wood fence that is perpendicular to the store
Thus, since area is 800 ft², then;
xy = 800
Length of fence; L = x + 2y
From earlier, xy = 800
y = 800/x
Thus;
L = x + 2(800/x)
L = x + 1600/x
Now, steel fencing costing $6 per foot and the other two sides will be constructed with wood fencing costing $3 per foot. Thus, total cost is;
C(x) = 6x + 3(2y)
But y = 800/x. Thus;
C(x) = 6x + 3(1600/x))
C(x) = 6x + 4800/x
C'(x) = 6 - 4800/x²
At C'(x) = 0, the cost is minimized.
Thus=
6 - 4800/x² = 0
6x² = 4800
x² = 4800/6
x² = 800
x = √800
x = 20√2
When 0 < x < 20√2, C'(x) < 0, so we say that C(x) is decreasing
When x > 20√2, C'(x) > 0, so we say that C(x) is increasing
Thus, the cost is minimized when x = 20√2
Thus, putting 20√2 for x in y = 800/x, we have;
y = 800/(20√2)
y = 40/√2
Thus, dimensions that will minimize cost are;
x = 20√2 ft and y = 40/√2 ft
