Optimization

Question

A fence must be built to enclose a rectangular area of 45000 square feet. Fencing materials costs $4 per foot for the two sides facing north and south and $8 per foot for the other two sides. Find the cost of the least expensive fence.

Answer 1

Let horizontal side of the fence = x

Let the vertical side of the fence = y

 

Now we set up equations for the area and cost.

 

xy = 45000         eq1      (Area)

 

2(4x) + 2(8y) = Cost       

 

8x + 16y = Cost            eq2

 

 

The least expensive fence means the minimum cost.  The less materials used, the less the cost .

 

If we substitute eq1 into eq2, we get

 

8x + 16(45000x^-1) = Cost

 

 

Use the first derivative test to find the minimum value of x.  Set the derivative of the function equal to zero.

 

d/dx (Cost) = 0

 

8 + 16(-45000x^-2) = 0

 

8 - 16(45000x^-2) = 0

 

 

Subtract 8 on both sides of equation.

 

-16(45000x^-2) = -8

 

 

Divide both sides of the equation by -16.

 

45000x^-2 = 1 / 2

 

45000 / x² = 1 / 2

 

 

Cross multiply.

 

x² = 90000

 

x = ±√(90000)

 

x = - 300        and        x = 300

 

 

These are your critical points.  They are the location of your minimum values.

 

Since the dimensions can only be positive, the only possible minimum value we can have is  x=300

 

The minimum sides are then 300 feet and 150 feet.

 

Total cost = 8(300) + 16(150)

               = 2400 + 2400

               = 4800

 

The least expensive fence costs $4800.