# Determine the maximum number of students that can fit in a room given the height and the width of the room.

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Given width of room say W and height H. Let us assume length of the room be L and room is in form of cubiod.

Define area of the floor as a function of L and W i.e.

`A(L,W)=L xx W`

and Volume of the room

`V=LxxWxxH`

If relation between L and W are known to us then A(x,y) can be transformed into a single variable function either in L or W .

Now let hypothetically assume that average area `S(x,y)` required for sitting an individual is known . Then we find min/max number of students with respect to sitting constraints i e. area of the floor of the room . If we know how much air reqired for each students ,then we can set constraints volume of the room.

Define Lagranges function

`F(L,W)=V(L,W,H)+lambda S(x,y)`

We can find relation between W and Y or X and W .

Define

`(delF)/(delL)=0`

`(delF)/(delx)=0`

`(delF)/(dellambda)=0`

Solve for `L,x and lambda` above system of equations.

By this method you can estimate min /max .