A rancher plans to set aside a rectangular region of one square kilometer for cattle and wishes to build a wooden fence to enclose the region. Since one side of the region will run along the road, the rancher decides to use a better quality wood for that side which costs three times as much as the wood for the other sides. What dimensions will minimize the cost of the fence?
To minimize the cost of the fence, we need to determine the dimensions of the rectangular region.
Let's assume the length of the rectangular region is 'x' and the width is 'y' (both in meters).
Given that one side of the region will run along the road and will use a better quality wood that costs three times as much as the wood for the other sides, we can divide the rectangular region into two parts:
1) Side parallel to the road (width y) using the better quality wood, and
2) The other three sides (two lengths of x and one width of y) using regular wood.
The cost of the fence will be the sum of the costs for both types of wood.
To calculate the cost, we need to determine the lengths of each side:
1) The length of the side parallel to the road (width y) is y.
2) The length of the other three sides (two lengths of x and one width of y) is 2x + y.
The cost of using the better quality wood for one side (width y) is three times the cost of using regular wood:
Cost_1 = 3y
The cost of using regular wood for the other three sides (lengths 2x + y) is (2x + y) times the regular cost of one unit of wood:
Cost_2 = (2x + y)
The total cost of the fence is the sum of Cost_1 and Cost_2:
Total Cost = 3y + (2x + y)
We know that the area of the rectangular region must be 1 square kilometer, which can be represented as:
Area = length x width
1 = x * y
We can express y as a function of x by rearranging the equation above:
y = 1/x
Substituting the expression for y in terms of x into the equation for the total cost, we have:
Total Cost = 3(1/x) + (2x + 1/x)
To find the dimensions that minimize the cost, we need to differentiate the total cost equation with respect to x, set it equal to zero, and solve for x. This will give us the value of x that minimizes the cost.
Differentiating Total Cost with respect to x:
d(Total Cost)/dx = d(3(1/x) + (2x + 1/x))/dx
= -3/x^2 + 2 - 1/x^2
= -4/x^2 + 2
Setting the derivative equal to zero:
-4/x^2 + 2 = 0
-4/x^2 = -2
-4 = -2x^2
x^2 = 2
x = √2
So the value of x that minimizes the cost is x = √2.
Substituting this value back into the equation for y, we can find the corresponding value of y:
y = 1/√2 = √2/2
Therefore, the dimensions that minimize the cost of the fence are:
Length (x) = √2
Width (y) = √2/2
The rancher should set aside a rectangular region with a length of √2 kilometers and a width of √2/2 kilometers to minimize the cost of the fence.
x y = 1
c = 3 x + x + 2 y = 4 x + 2 y
c = 4 x + 2/x
dc/dx = 4 - 2/x^2
= 0 for minimum
4 = 2/x^2
x^2 = 1/2
x = 1/sqrt 2 = sqrt 2/2 along road
y = sqrt 2