Calculate the width and length of the fields to minimise the total amount of fencing used  A farmer wishes to build a large rectangular grazing field of dimensions width and length, splitting the...

Calculate the width and length of the fields to minimise the total amount of fencing used

 

A farmer wishes to build a large rectangular grazing field of dimensions width and length, splitting the area into 5 equal sub-fields by dividing the width into 5 equal pieces. The total area of the field must be 11500 square feet. Find the dimensions of the 5 equal sub-fields so that the least amount of fencing is used. 

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mathsworkmusic | (Level 2) Educator

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Let `A` be the total area of the field, `l` be the length of each sub-field and `w` be the width of each sub-field. Let `F` be the total amount of fencing.

Then,

`A = 5lw`  (the length of the field is `l` and the total width is `5w`)

`F = 6l +10w` (only one length of fencing is needed to divide two sub-fields)

We use the method of lagrange multipliers to minimise `F` with respect to the constraint `A = 11500` square feet

This involves simultaneously solving

1)  `(delF)/(dell) - lambda(delA)/(dell) = 0`

2)  `(delF)/(delw) - lambda(delA)/(delw) = 0`

where  `5lw = 11500`

`implies` simultaneously solving

1) `6 - 5lambdaw = 0`  2)  `10 - 5lambdal = 0`

This gives 1) `w = 6/(5lambda)`  2) `l = 2/lambda`

which leads to the constraint `12/lambda^2 = 11500` `implies`  `lambda = +-sqrt(12/11500)`

`lambda` must be positive so we have

`l = 2sqrt(11500/12) = 61.914`ft , `w = 6/5sqrt(11500/12) = 37.148` ft 

 

L = 61.914 ft

W = 37.148 ft

 

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