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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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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