Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.

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### Take the dimensions of a rectangular box are.

**Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.**. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. Write down the equation of a sphere in standard form and then write an equation for the volume of the rectangular box. And let x y and z be the x- y- and z-coordinates of the corner of a rectangular prism which is inscribed in this sphere.

Were asked to find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere. Then the volume of the rectangular box is V 8xyz where the dimensions are 2 x 2 y and 2 z. My Partial Derivatives course.

An inscribed rectangular box is uniquely determined 1 by the xy-coordinate of its corner in the first octant so we can compute the z coordinate of this corner by x2y2z2r2 z r2x2y2. What Id like to ask is how should I set up my functions to solve this using Lagrange. Let the equation of a sphere x2 y2 z2 r2 where x y z are parameters and r is the radius of the sphere.

The maximum rectangular box that can be inscribed in a sphere of radius R is a cube whose side is R 2 therefor the maximun volume of that cube is R 2 R 2 R 2 R3 2 32. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. For the derivative tests method assume that the sphere is centered at the origin and consider the circular projection of the sphere onto the xy-plane.

The diagonal of the cube inside the sphere is equal to the diameter of the sphere or twice the radius. Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 147. The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is View solution A solid iron rectangular block of dimensions 4.

6 m and 1 m is cast into a hollow cylindrical pipe of internal radius 3 0 cm and thickness 5 cm. The length of the diagonal is. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.

This video shows how to find a right circular cylinder with largest volume that can be inscribed in a sphere of radius r. 4 m 2. Let the largest rectangular box will be a cube.

Let be the fixed radius of a sphere centred at the origin. The diameter of the sphere is the space diagonal of the cube. Calculus MindTap Course List 8th Edition.

Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. Find the length of the pipe.

In this video Krista King from integralCALC Academy shows how to find the largest possible volume of a rectangular box inscribed in a sphere of radius r. The problem is find the maximum volume of our rectangular box that is inscribed in your spear of writers are here within notice The dimensions of the box A sex Why sing Place The box is corrupt in a sphere of writers are so we have to or in relation X square plus y square us. Thus the maximum value of the function Vx y z 8xyz subject to the equation x2 y2 z2 r2.

The maximum volume of any rectangular box would be a cube. Now The volume V of the rectangular prism is. More_vert Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 147.

I can get you started by telling you how the solutions manual got the volume as V8xyz. The space diagonal of a cube is sqrt 3 times its side length. So now we have to find the volume of a cube inscribed in a sphere.

The volume of the box inscribed inside a sphere where x y z is the point in octant 1 where the box.

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