Determine the volume of the largest box in the first octant with three in the coordinate planes, one vertex at the origin, and its opposite vertex in the plane x+3y+5z=15

Alyce Wilkinson 2021-02-25 Answered

Determine the volume of the largest box in the first octant with three in the coordinate planes, one vertex at the origin, and its opposite vertex in the plane x+3y+5z=15

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funblogC
Answered 2021-02-26 Author has 91 answers

The volume of the largest rectangular box is to be determined based on the first octant with three faces in the coordinate planes, and one vertex in the plane x+3y+5z=15.
The volume of the rectangular box that lies in the first octant based on the three faces that lies in the coordinate plane as follows,
V=f(x,y)=xyz
The vertex get lies in the plane as below,
x+3y+5z=15
5z=15x3y
z=15x3y5
The volume is determined as below,
V=f(x,y)=15xy(15x3y)=3xyx2y53xy25
The Volume will be maximum if fx=fy=0
As f(x)=3y25xy35y2=15y(152x3y)=0
y=0, y=152x3 (1)
And
fy=3xx256xy5=15x(15x6y)=0
x=0 and substituting y=0
x=15
At y=152x3
fy=0
15x(15x6152x3)=0
x=0,15x30+4x=0
3x=15
x=5
y=15203=153
At x=0
y=5
And at x=5, y=53
The Critical point are (0,0),(15,0),(0,5),(5,53) and z
At (5,53)isz=15x3y5
z=15555
z=1
The Volume of the longest rectangular box is determined as below,
553=253
Hence, the volume is 253

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