At the point of the least cost of fencing the cost function has a zero
derivative.
Reasons:
Shape of the pen = Rectangular
Area of the pen = 24 yd²
Cost of material on one side of the fence = $6 per yard
Cost of material on the remaining three sides = $3 per yard
Required:
The least cost of fencing the pen
Solution:
The least cost is a minimum value of the cost function
Let L represent the length of the fence and let W represent the width of the fence
We have;
The perimeter of the fence = 2·L + 2·W
The area of the pen, A = L × W = 24
One of the length, L, of the fence costs $6 per yard and the other length L,
of the opposite side costs $3 per yard.
The cost of fencing, C 3 × 2·W + 3 × L + 6 × L = 6·W + 9·L
C = 6·W + 9·L
[tex]\displaystyle W = \mathbf{\frac{24}{L}}[/tex]
Which gives;
[tex]\displaystyle C = 6 \cdot \frac{24}{L} + 9 \cdot L = \frac{144}{L} + 9 \cdot L = \mathbf{\frac{144 + 9 \cdot L^2}{L}}[/tex]
The shape of the above function is concave upwards.
At the minimum value, we have;
[tex]\displaystyle \frac{dC_{min}}{dL} = \frac{d}{dL} \left(\frac{144}{L} + 9 \cdot L\right) = 9 - \frac{144}{L^2} = 0[/tex]
Which gives;
[tex]\displaystyle \frac{144}{L^2} = 9[/tex]
[tex]\displaystyle \frac{144}{9} = L^2[/tex]
By symmetric property, we have;
[tex]\displaystyle L^2 = \frac{144}{9}[/tex]
[tex]\displaystyle L = \sqrt{ \frac{144}{9}} = \frac{12}{3} = 4[/tex]
The length of the fence that gives the least cost, L = 4 yards
[tex]\displaystyle At \ L = 4, \ W = \frac{24}{4} = 6[/tex]
The width of the fence at least cost, W = 6 yards
Cost, C = 6·W + 9·L
Least cost, [tex]C_{min}[/tex] = 6 × 6 + 9 × 4 = 72
The least cost of the fencing, [tex]C_{min}[/tex] = $72.00
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