Answer:
the lengths of the sides of the rectangular field should be:
x = 3,000√2 ft
y = 1,000√2 ft
to minimize the cost of the fence.
Explanation:
The rectangular field can be divided into two equal parts with a fence parallel to one of the sides of the rectangle. So, each part of the field will have an area of 3 million square feet.
Let x be the length (in feet) of a side parallel to the dividing fence, and y be the length (in feet) of a side perpendicular to the dividing fence. Then, we have:
xy = 6,000,000/2 = 3,000,000
Solving for y, we get:
y = 3,000,000/x
The amount of fencing needed is:
F = 2x + 3y
Substituting y = 3,000,000/x, we get:
F = 2x + 9,000,000/x
To find the critical numbers, we take the derivative of F with respect to x:
F' = 2 - 9,000,000/x^2
Setting F' equal to zero, we get:
2 - 9,000,000/x^2 = 0
Solving for x, we get:
x = ±√(9,000,000/2) = ±3,000√2
Since x cannot be negative, the only critical number is:
x = 3,000√2
To find the minimum value of F, we need to check the endpoints of the possible values of x. Since we cannot have a rectangle with one side of length zero, the endpoints are when x approaches infinity and when x approaches zero. As x gets very large, the value of F also gets very large. As x approaches zero, the value of F approaches infinity. Therefore, the minimum value of F occurs at the critical number:
x = 3,000√2
Substituting this value of x into the equation for y, we get:
y = 3,000,000/x = 3,000,000/(3,000√2) = 1,000√2
So, the lengths of the sides of the rectangular field should be:
x = 3,000√2 ft
y = 1,000√2 ft
to minimize the cost of the fence.