Question

A package to be mailed using a certain postal service may not measure more than 84 inches in length plus girth. (Length is the longest dimension and girth is the largest distance around the package, perpendicular to the length.) Use Lagrange multipliers to find the exact dimensions (in inches) of the cylindrical mailing tube of greatest volume that may be mailed using the postal service.

Answer 1

To find the dimensions of a cylindrical mailing tube with maximized volume under a length plus girth constraint, we employ the method of Lagrange multipliers. The final solution yields a tube with a radius of 12 inches and a height of 36 inches, meeting the postal service's size restrictions.

Step-by-step explanation:

When using Lagrange multipliers to find the dimensions of a cylindrical mailing tube of greatest volume subject to a length plus girth restriction, we are dealing with a problem of constrained optimization. We want to maximize the volume of a cylindrical package (V = π * r^2 * h) subject to the constraint that the total length plus girth is equal to or less than 84 inches (L + 2πr ≤ 84).

The Lagrange function L we will look at is L = π * r^2 * h - λ * (2π * r + h - 84), where λ is our Lagrange multiplier. Setting the partial derivatives equal to zero, we have the following system of equations:

1. 
   
3. ∂L/∂r = 2πrh - λ * (2π) = 0
4. 
   
6. ∂L/∂h = πr^2 - λ = 0
7. 
   
9. ∂L/∂λ = 2π * r + h - 84 = 0

Solving this system, we get that r = 12 inches and h = 36 inches for the mailing tube that maximizes volume within the given constraints.