Work only the ones that are circled thank you so much Find the maximum volume of

Question

Work only the ones that are circled thank you so much Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. Find the dimensions of the box with volume 1000 cm^3 that has minimal surface area. 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 + 2y + 3z = 6. Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm^2. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.

Explanation / Answer

So, we will maximize V= (62y3z)yz on the open set y>0 and z>0.The maximum of V will be attained at one of its critical points. To find the critical points simplify to
V= 6yz2y2z3yz^2 so
Vy= 6z4yz3z^2=z(64y3z)
Vz= 6y2y^26yz= 2y(3y3z)
Since y and z are not 0, the critical point equations Vy=0 and Vz=0 are equivalent to the following system of linear equation
4y+3z=6
y+3z=3
Subtracting the second equation from the first we see that 3y=3 so y=1. from this it follow that z=2/3, and x=6-2y-3z=2.
the maximum volume is 2*1*(2/3)=4/3

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