# The perimeter of a room is 400 meters. Find the number of maximum number of students in a room if an average student occupies 2.5 meters of space. Please use derivatives

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Let us say room has length x ,width w and area A.

Perimeter `= 2(x+w)`

`400 = 2(x+w)`

`w = 200-x`

It is given that a student occupies 2.5m^2 of space. When the area of the room maximizes then the student will also maximize.

If the amount of students are P then;

`P = A/2.5`

`P = (x*w)/2.5`

`P = (x*(200-x))/2.5`

`P = (200x-x^2)/2.5`

For maximum or minimum number of students` (dP)/dx = 0`

`(dP)/dx = 1/2.5(200-2x)`

When `dP/dx = 0`

`1/2.5(200-2x) = 0`

`x = 100`

If P has a maximum at x = 100 then `[(d^2P)/dx^2]_(x = 100)` is a negative value.

`(d^2P)/dx^2 = -2<0`

So we have a maximum for P.

Maximum area of the room `= 100x100m^2`

Maximum students `= (100xx100)/2.5 = 4000`

*So we can have maximum 4000 students in the room.*

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