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**

## We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support

Already a member? Log in here.

Are you a teacher? Sign up now