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#?

Answer 1

#V = x_0y_0z_0=400/3#

This problem can be stated as a maximum one.

Find #max f(x,y,z)=xyz# subject to #g(x,y,z) = x + 6 y + 10 z - 60#

The lagrangian is

#L(x,y,z,lambda)= f(x,y,z)+lambda g(x,y,z)#
#L# is analytic and the stationary points include the minima/maxima points.

The stationary points are the solutions of

#grad L(x,y,z,lambda)= vec 0#

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

#f@g= (x y)/10 (60 - x - 6 y)#

determining the eigenvalues of

#H=grad^2 f@g = ((-y/5, 6 - x/5 - (6 y)/5),(6 - x/5 - (6 y)/5, -(6 x)/5))#
The local maxima are the stationary points in which #H -< 0# with negative eigenvalues. The maximum point is
# (x= 20, y= 10/3, z = 2)#
with eigenvalues #{-24.1702,-0.496479}# and with volume
#x_0y_0z_0=400/3#
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Answer 2

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:

  1. Determine the equation for the volume of a rectangular box: ( V = l \times w \times h ).
  2. Recognize that the box's dimensions are parallel to the coordinate axes, so its sides are parallel to the coordinate planes.
  3. 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 ).
  4. 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.
  5. 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.
  6. Solve the system of equations formed by these constraints to find the values of ( x ), ( y ), and ( z ).
  7. 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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Answer from HIX Tutor

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.

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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