Question

How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, and one vertex in the plane x+7y+11z=77?

Answer 1

the volume is `5929/27 approx 219.6 " cubic units"`

Explanation:

you're optimising volume function

`f(x,y,z) = xyz`

with constraint `g(x,y,z) = x + 7y + 11z = 77`

using the Lagrange Multiplier ie `nabla f = lambda nabla g`

we get

`[(yz),(xz),(xy)] = lambda [(1),(7),(11)]`

comparing `lambda`'s and from rows 1 and 2

`(yz)/1 = (xz)/7 implies x = 7y`

from rows 2 and 3

`(xz)/7 = (xy)/11 implies z = (7y)/11`

if we park those back into the constraint

`7y + 7y + 11( (7y)/11 ) = 77`

`implies y = 77/21`

so `x = 77/3, z = 7/3`

So the volume is `77/3 * 77/21 * 7/3 = 5929/27 approx 219.6 " cubic units"`

in terms of scoping for a reality check, the total volume under that plane in the first octant is

`\int_(x=0)^(77) quad \int_(y=0)^( 11 - x/7) quad (77-x-7y)/11 quad dy \ dx approx 988 " cubic units"`