Find the dimensions of the rectangular box of maximum volume that has three of its faces in the coordinate planes, one vertex at the origin, and another vertex in the first octant on the plane 4 3 12 x y z    .

Q: Find the dimensions of the box with square base, surface area 20, and maximal volume.

A: A box with square base has surface area 20we have to find its dimension and maximum volume.

Q: 4. Find the volume of the wedge cut from the first octant by the cylinder z = 12 – 3y², and the…

A:

Q: An open top box is to be made from 48 m of wood with dimensions x, y and z as shown in figure. Using…

A:

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A: The given equation of plane is : x+2y+3z=6 x=6-2y-3z                   …(1) Volume of a rectangular…

Q: Find the dimensions of the rectangular box of maximum volume that has three of its faces in the…

A:

Q: 9 A copper sheet with 16 cm by 16 cm dimension has to be made as a square box with no top cover.…

A: Dimension of the copper sheet = 16 by 16 cm;                    Square corners…

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A:

Q: Find the plane x/a + y/b + z/c = 1 that pass-es through the point (2, 1, 2) and cuts off the least…

A:

Q: Find the minimum and maximum squared distance between the sphere x² + y² + z² = 9 and the point…

A: We have to find the minimum and maximum squared distances between the sphere x2+y2+z2=9 and the…

Q: 2. Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on…

A: An equilateral triangle can be inscribed in the epitrochoid and that its centroid is onthe circle of…

Q: Consider an axis-aligned cuboid (i.e. with one vertex at the origin and with the 3 edges connected…

A:

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A:

Q: 4. Find the volume of the wedge cut from the first octant by the cylinder z = 12 - 3y2, and the…

A: We have to find volume.

Q: Show that a closed right circular cylinder of given surface has maximum volume if its height equals…

A: Let, the surface area be S, radius be r, height be h and volume be V. The total surface area be…

Q: Find the dimensions of a cylindrical can with a bottom but no top of volume 4 m3 that uses the least…

A: To Find: Dimension of a cylindrical can with a bottom but no top of volume 4 m3that uses the least…

Q: Consider a rectangular box B that has a bottom and sides but no top and has minimal surface area…

A:

Q: Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in…

A: According to our company policy, I am supposed to answer only one question at a time so for the rest…

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A: We have to find the volume of the largest rectangular box in the first octant with three faces in…

Q: A box manufacturer wishes to construct a closed box of volume 288 in ^3 where the base is a…

A:

Q: find the dimension of a cylinder that can be inscribed in a given sphere if it's convex surface area…

A:

Q: Find the area of the region cut from the plane x+4y+8z=9 by the cylinder whose walls are x=y2 and…

A: Given, x+4y+8z=9 x=y² x=18−y² Therefore, The surface is  f=0 with f= x+4y+8z-9

Q: Find the dimensions of the box of maximum volume with its sides parallel to the coordinate planes…

A: Given :                    The ellipsoid is :                           xa2 + yb2 + zc2 = 1…

Q: Find the maximum volume of a cylindrical can such that the sum ofits height and its circumference is…

A: Let r denotes radius       h denotes height Then       volume of cylinder (V) = πr2h Restriction is…

Q: Show that the maximum value of a2b2c2 on a sphere of radius r centered at the origin of a Cartesian…

A: Let fa,b,c=a2b2c2 subject to a2+b2+c2 =r2  The gradient equation is…

Q: Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its…

A:

Q: a

A:

Q: Calculate the volume under the elliptic paraboloid z = 4x² + 6y2 and over the rectangle R = [-2, 2]…

A:

Q: Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the…

A: Let 2x, 2y, 2z be the dimensions of rectangular box. Then Let

Q: A closed rectangular box with faces parallel to the coordinate planes has one bottom corner at the…

A: Let x , y, z donate  the length, width and the height of the rectangular box.  Volume of the box is…

Q: A rectangular box has three of its faces on the coordinate planes and one vertex in the first octant…

A: Given,The box's volume is  V = xyz  but we know that   z = 4 − x2 − y2V = xyz    = xy ( 4 − x2 − y2…

Q: Show that the rectangular box of maximum volume that can be inscribed in the sphere x^2 + y^2 + z^2…

A:

Q: Calculate the volume under the elliptic paraboloid z = 4x2 + 6y2 and over the rectangle R = [-2, 2]…

A: The solution for the above question is shown below.

Q: Find the dimensions of the box with square base with: (a) Volume 12 and the minimal surface area.…

A:

Q: 7. Show that a closed rectangular box of maximum volume having a prescribed surface area S is a…

A: Let the volume function be V=lbh and the surface area is 2lb+bh+lh=S. Take the constraint to be gl,…

Q: Find the dimensions of the rectangular box of maximum volume that can be inscribed inside the sphere…

A: Given sphere is

Q: Use Lagrange multipliers to find the dimensions of a box with volume 1000 cm3 that has minimal…

A: the surface area of the box is  S=2xy+2yz+2xz................(1) the volume of the box is given by…

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A:

Q: A rectangular box with no top is to have a surface area of 16 m2. Find the dimensions (in m) that…

A: Use derivative test for calculating the maximum volume of a rectangular box. Assume that base of box…

Q: Find the dimensions of the circular cylinder of maximum volume that can be inscribed in a sphere of…

A:

Q: 6. Show that when a hole of radius a is bored through the center of the sphere of radius b, the…

A: Given: The radius of the sphere is b. The radius of the hole is a. We have to show that the volume…

Q: Consider the region in the first quadrant bounded by y=x^2, the y-axis, and y=4. -the y-axis using…

A:

Q: 3. Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the…

A: Given

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A:

Q: 4. Four identical bricks are stacked up on top of each other. Each has a length of a and a mass of…

A:

Q: \ Use optimization to find the maximum volume of a box with one corner at the origin and the…

A: Assume, V be the volume function representing the volume of the box. l, b, h be the dimensions of…

Q: Show that the maximum value of abc on a sphere of radius r centered at the origin of a Cartesian…

A:

Q: A right circular cylindrical container with volume 167 cm³ is to be made. Use the method of Lagrange…

A:

Q: Find the volume of the largest rectangular box in the first octant with three faces in the…

A: