Thread: Optimization Problem

A builder must completely fence in 3 adjacent rectangular lots along a roadway. Each lot must have an area of 675 square metres. Fencing along the roadway costs $25/metre while side and back fencing costs $10/metre. Find the dimensions of each lot that minimize the total cost of fencing.

Hello, NAPA55!

The answers come out rather ugly . . .

A builder must completely fence in 3 adjacent rectangular lots along a roadway.
Each lot must have an area of 675 square metres.
Fencing along the roadway costs $25/metre while side and back fencing costs $10/metre.
Find the dimensions of each lot that minimize the total cost of fencing.

Code:

           x         x         x
      * = = = = * = = = = * = = = = *
      |         |         |         |
      |         |         |         |
     y|        y|         |y        |y
      |         |         |         |
      |         |         |         |
      * - - - - * - - - - * - - - - *
           x         x         x

There are m of fencing along the road.
At $25/m, its cost is: . dollars.

There are m of back fencing and m of side fencing.
At $10/m, its cost is: . dollars.

The total cost is: . dollars. .[1]


Each lot has an area of 675 m²: .

Substitute into [1]: .

And that is the function we must minimize . . .

Just to finish off what Soroban started...

Note: Remember your domain

, set the derivative equal to zero...

. (x cannot be negative, as denoted by our domain, therefore we take the positive).

Sub x back in and you should get

Therefore the dimension of each lot must be x to minimize the cost of fencing.