A rectangular garden of area exactly is to be enclosed on all four sides by fencing

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch Multipliers, non-holonomic constraints and forces outside the potential

Problem

A rectangular garden of area exactly

36\,\mathrm{m^2}

is to be enclosed on all four sides by fencing. Let

x

and

y

be the side lengths, so the constraint is

g(x,y) = xy - 36 = 0

and the quantity to minimise is the total fencing

P(x,y) = 2x + 2y

. ((a)) Use a Lagrange multiplier (

\nabla P = \lambda_1 \nabla g

) to find the dimensions that minimise

P

. ((b)) Give the value of

\lambda_1

at the minimum, with its unit.