1. Let a, b and c be 1, 1 and 9 respectively Consider the ellipsoid & given by the equations + = 1. a² 62 c² (a) Let P₁ = (±x, ±y, ±z) be the vertices of a rectangle inscribed in E. Explain why the volume of & is given by V = 8xyz. (b) Use the method of Lagrange multipliers to determine the largest possible volume of E.

Please help me with this homework. Thanks. (Please I need it to be systematic so that I can understand it thank you.)

The lateral surface of a water bottle consists of a circular cylinder of radius 3 and height 6, topped off by a truncated hyperboloid of one sheet of height 4. Assume the top of the truncated hyperboloid has a O 10 1 radius of 5. Find two equations that, when graphed together, form the lateral surface of the bottle. 6. Determine the correct equations. Select all that apply. O A. x2 y2 - 5(z - 10)2 = 1 O B. x2 - y? = 9 O C. 9x2 + 9y? - 5(z – 10)2 = 1 O D. x² + y² = 9 O E. 9x2 + 9y2 = 1 O F. - 9x2 – 9y² + 5(z – 10)2 = 1 O G. x2 + y? - z² = 9 O1. x? + y? + z? =9 O H. x² + y² - (z – 10)2 = 1 O J. 9x2 + 9y² + 5(z – 10)2 = 1

2. A manufacturer needs to make a cylindrical can that uses 100 square centimeters of material. Determine the radius of the can that will maximize the volume of the can. For a cylinder, V πr?h and Α= 2πrh+ 2πr? = T