a large closed storage rectangular box with a base constructed with two different types of wood. the base is made up of wood $5 per square feet and the top and sides are made of wood $3 per square feet suppose the amount available to spend is $1000 find the dimensions of the box with greatest volume

Answer 1

we want to maximize xyz subject to the constraint that

5xy+3(xy+2xz+2yz) = 1000

Using Lagrange multipliers, we find that x=y, and λ=2.1516 and z=4λ

That means a local max occurs at
(6.455,6.455,8.607)

A nice article on Lagrange multipliers, with an example similar to this problem, is at

http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx