Please answer the following questions # 10, 12, 13, 18, 24.

Do exercises # 10, 12, 13, following the same exact format from the examples that are going to be provided.

7. Maximize Q = xy , where x and y are positive numbers, such that x + yz = 1. 8. Maximize Q = xy, where x and y are positive numbers, such that x + y2 = 4. 9. Minimize Q = 2x2 + 3y', where x + y = 5. 10. Minimize Q = x2 + 2y2, where x + y = 3. 11. Minimize Q = x2 + y', where x + y = 20. 12. Minimize Q = x2 + y, where x + y = 10. 13. Maximize Q = xy, where x and y are positive num- bers, such that x'+ y = 16. 14. Maximize Q = xy, where x and y are positive num- bers, such that x + =yz = 1. 15. Maximize Q = Vx + Vy, where x + y = 1. 16. Maximize Q = Vx + Vy, where 2x + y = 6.18. Maximizing Area. A rancher wants to enclose two rectangular areas near a river, one for sheep and one for cattle. There are 240 yd of fencing available. What is the largest total area that can be enclosed? River X X X 240- 3x 19. Maximizing Ared. A carpenter is building a rectan- gular room with a fixed perimeter of 54 ft. What are the dimensions of the largest room that can be built? What is its area? 20. Maximizing Area. Of all rectangles that have a perimeter of 34 ft, find the dimensions of the one with the largest area. What is its area? 21. Maximizing Volume. From a thin piece of cardboard 30 in. by 30 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? 22. Maximizing Volume. From a thin piece of cardboard 20 in. by 20 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? 23. Minimizing Surface Area. A container company is designing an open-top, square-based, rectangular box that will have a volume of 62.5 in'. What dimensions yield the minimum surface area? What is the minimum surface area? 24. Minimizing Surface Area. A soup company is con- structing an open-top, square-based, rectangular metal tank that will have a volume of 32 ft. What dimensions yield the minimum surface area? What is the minimum surface area?\fDate No YO , X=1- 02 = 1 Y = J z . x = 1- ( 1 4 )= 2 = X 8 Maximize Q=xy z , x + y z = 4 X+ W/ 2 = 4 1/ 2 - yz Check Q Z - X = 4- 42 plug y = 0 ; x = 4 - 02 = 4 n this Q = 4 (0 )2 = = can't be bc one Q = ( 41- yz) yz is O = 4 42 - yu y = vz / x = 4 - ( 12 ) 2 = 121 da = 8y - 4 43 G = 2 ( VZ ) 2 = 14 vaiwe of This problem have X 8 y - 41 13 = 0 3 Solutions y y ( 2 - 4 2 ) = 0 LY = 0 2 - 42=0 4 = 0 Ty 2 = 2 y = + 12 y = - 12 reject morning gloryDate No (1 ) Minimize Q = 2 x 2 + 3 y2 , Xty = 5 X + y = s - ? 1 = 5 - x -7 plug in derivative (Q = 2 x 2 + 3 ( 5 - x ) 2 7 ? da = 4 x + 3 ( 2 ) ( 5 - x ) ( -1 ) - 7 " chain ax 4 x + (6 ) (5 - X 1=10 Bute 4 x - 30+ 6X = 0 10 x - 30 = 0 +360 + 30 10 X = 30 10 X = 3 1 = 5 - 3 = 12= 4 = 2 ( 3 ) 2 4 3 ( 2 ) 2 = 30 1 1 ) minimize Q = * 2 ty z xty = 20 Xty = 20 * = 20- 4 0 = ( 20 - 4 1 2 + y z y=1O X = 40 = 2 (20 - 4) ( - 1) + 2 y Q 2 = 102+ 102 = 200 = - 40 +24 +2 y = 44 y-40