Calling a,b,c the dimensions we have a restriction
a b c = 324
and an objective function
f(a,b,c) = c_1(2(ab+ac)+bc)+c_2bc
where c_1,c_2 are the construction costs with the relationship
c_2 = 2c_1 or
f(a,b,c) = c_1((2(ab+ac)+bc)+2bc)
This problem can be handled using the so called Lagrange multipliers
The lagrangian
L(a,b,c,lambda) = f(a,b,c)+lambda(abc-324)
The stationary points are the solutions to
grad L = vec 0
where grad=(partial/(partial a),partial/(partial b),partial/(partial c),partial/(partial lambda)) so we have
(L_a,L_b,L_c,L_(lambda))=(0,0,0,0) or
{(2 (b + c) c_1 + b c lambda=0),( (2 a + 3 c) c_1 +
a c lambda=0), ((2 a + 3 b) c_1 + a b lambda=0), ( a b c-324=0):}
Solving for a,b,c,lambda we obtain
a=9,b=6,c=6,lambda=-2c_1/3
