Lagrange Multipliers for Volume

[stardusto12]

Homework Statement

The Park Service is building shelters for hikers along the Appalachian Trail. Each shelter has a back, a top, and two sides. Find the dimensions that will maximize the volume while using 384 square feet of wood.

They want me to find the length, width, and height.

Homework Equations

Here's my problem, I can't form a constraint for the problem because I don't know what they are looking for. The constraints I formed (two of them) did not work (they are below along with the formula I used).

F(x)=2zy+xz+xy
constraint used= Lambda(384-xyz)

and

f(x)=xyz
constraint used= lambda(384-2yz-xz-zy)

The Attempt at a Solution

The answers I got are WAY off the correct answers. The correct answers are 16 (length) and 8 (for both width and height).

Can someone please tell me if I used the wrong equations, and if I did, how to form an equation from the given problem. I know how to calculate the answers, just not how to form the equation from a word problem. Thank you.

 

[Dick]

The second form is correct. Extremize xyz-lamba(384-2yz-xz-xy) (note the typo in the what you have shown - xy turned into zy). The constraint is on area, not volume.

 

[HallsofIvy]

By the way, the Park Service does not build shelters along the Appalachian trail. They are built entirely by volunteers, members of the Appalachian Trail Conservancy and its affiliates.