# 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 + 6y + 10z = 60#?

This problem can be stated as a maximum one.

The lagrangian is

The stationary points are the solutions of

or

#{ (lambda + y z =0), (6 lambda + x z = 0), (10 lambda + x y = 0), (-60 + x + 6 y + 10 z = 0) :}#

with solutions

#( (x = 0, y= 0, z= 6, lambda = 0), (x= 0, y= 10, z= 0, lambda = 0), (x= 20, y= 10/3, z = 2, lambda= -20/3), (x= 60, y= 0, z= 0, lambda=0) )#

The solutions are in the restriction manifold so them will be qualified on

determining the eigenvalues of

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To 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 + 6y + 10z = 60 ), you can follow these steps:

- Determine the equation for the volume of a rectangular box: ( V = l \times w \times h ).
- Recognize that the box's dimensions are parallel to the coordinate axes, so its sides are parallel to the coordinate planes.
- Identify the constraints for the box: it must have three faces in the coordinate planes (x-y plane, y-z plane, x-z plane), and one vertex in the plane ( x + 6y + 10z = 60 ).
- Set up the equations for the dimensions of the box:
- Let ( x ) represent the length along the x-axis.
- Let ( y ) represent the length along the y-axis.
- Let ( z ) represent the length along the z-axis.

- Apply the constraints to these dimensions:
- The vertex in the plane ( x + 6y + 10z = 60 ) means that ( x + 6y + 10z = 60 ) is satisfied.
- The three faces in the coordinate planes mean that ( x = 0 ), ( y = 0 ), and ( z = 0 ) are satisfied.

- Solve the system of equations formed by these constraints to find the values of ( x ), ( y ), and ( z ).
- Once you have the values for ( x ), ( y ), and ( z ), plug them into the volume formula ( V = l \times w \times h ) to calculate the volume of the rectangular box.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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