A farmer wants to fence a rectangular area of 800,000 m² and divide it in half with a fence that is parallel to one of the sides of the rectangle, and twice as expensive as the fence on the outer sides. How can this be done in order to minimize the cost of the fence?
Hint: If all the fencing costs the same, what are we needing to minimize in this scenario?
To minimize the cost of the fence, we need to find the dimensions of the rectangular area that will result in the minimum length of fencing. Since we are given that the area of the rectangle is 800,000 m², we can express this as follows:
Area = Length × Width = 800,000 m²
Now, we need to divide the rectangle into two equal halves with a fence that is parallel to one of the sides of the rectangle. Let's assume that the width of the rectangle is divided by the fence. In that case, the width of one half would be Width/2, and the width of the other half would also be Width/2.
The perimeter of each half would be:
Perimeter of First Half = Length + 2 × (Width/2) = Length + Width
Perimeter of Second Half = Length + 2 × (Width/2) = Length + Width
The total cost of the fence is the sum of the two perimeters, where the outer fence costs a certain amount per meter and the inner fence (which divides the rectangle) costs twice that amount per meter. Let's say the cost per meter of the outer fence is 'c', then the cost per meter of the inner fence would be '2c'.
Total Cost = (Length + Width) × c + (Length + Width) × 2c
= 3 × (Length + Width) × c
Since we need to minimize the cost of the fence, we need to minimize the term (Length + Width). However, we can't directly minimize their sum. So let's express one variable in terms of the other to eliminate one variable:
Length × Width = 800,000 m² (Equation 1)
From Equation 1, we can express Length in terms of Width as:
Length = 800,000 m² / Width
Substituting Length in terms of Width in the total cost equation:
Total Cost = 3 × (800,000 m² / Width + Width) × c
Now, we have the cost in terms of a single variable, Width. To find the minimum cost, we can take the derivative of the cost equation with respect to Width and set it equal to zero:
d(Total Cost)/d(Width) = 0
By solving this equation, we can find the value of Width that minimizes the cost. Substituting this value of Width back into the equation for Length will give us the corresponding value of Length.
maybe it's time you start including some of your work in these posts. You should have some ideas by now...
2xy = 800000, so y = 400000/x
The cost
c(x) = 2y + 2(4x+2y) = 8x+6y = 8(x+300000/x)
c'(x) = 8(1-300000/x^2)
Now just find when c'=0