Week 2_Abiodun Macfoy_ Business Calculus Series: Deriving Simple Cost Equation

Now consider this simple real-life application that was asked by one of my students as an instructor and to the class in general.

Question: A property owner intends to fence a garden close to the road. The fencing next to the road should be sturdier and it cost N500 per foot while the other fencing cost N300 per foot. The garden has an area of 1200 sq/ft

1. Derive the equation that represents the cost of fencing the garden
   
2. Find the dimensions that minimizes the cost of fencing
   

Solution:

Part A.

Let represent length and represent width. Therefore:

Cost next to the road

Cost of the side opposite the road

Cost of width; since there are 2 sides of the width, it will be represented twice.

 

Hence the equation becomes:

(1)

 

 

Part B.

Area

 

If we substitute the above into equation (1) above, the problem will be to minimize. Then we have:

 

                        (2)

 

 

Equation (2) can be graphed and traced to determine that will the cost of fencing or we can differentiate equation (2) and solve for

If we differentiate equation (2), we'll have:

 

 

 

Hence

To find that will minimize cost, we substitute into , therefore . The solution to minimize cost has assigned a smaller ft size to the costlier side of the fence which is 30ft.

The length and width that will minimize cost is

 

 

Reference:

Calculus for Business, Economics, and Social Science and Life Science

Laurence D. Hoffman, Gerald Bradley

 

Intermediate Microeconomics: A Modern Approach

Hal R. Varian