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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aruv | High School Teacher | (Level 2) Valedictorian

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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 .

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