# 11X1 T12 06 maxima & minima (2010)

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1. Maxima & Minima Problems 2. Maxima & MinimaProblems When solving word problems we need to; 3. Maxima & Minima Problems When solving word problems we need to; (1) Reduce the problem to a set of equations (there will usually be two) 4. Maxima & MinimaProblems When solving word problems we need to; (1) Reduce the problem to a set of equations (there will usually be two)a) one will be what you are maximising/minimising 5. Maxima & MinimaProblems When solving word problems we need to; (1) Reduce the problem to a set of equations (there will usually be two)a) one will be what you are maximising/minimisingb) one will be some information given in the problem 6. Maxima & MinimaProblems When solving word problems we need to; (1) Reduce the problem to a set of equations (there will usually be two)a) one will be what you are maximising/minimisingb) one will be some information given in the problem (2) Rewrite the equation we are trying to minimise/maximise with only one variable 7. Maxima & MinimaProblems When solving word problems we need to; (1) Reduce the problem to a set of equations (there will usually be two)a) one will be what you are maximising/minimisingb) one will be some information given in the problem (2) Rewrite the equation we are trying to minimise/maximise with only one variable (3) Use calculus to solve the problem 8. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. 9. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. 10. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. ss 11. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s 12. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs1 13. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs16s 2r 362 14. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs1 6s 2r 362make r the subject in 2 15. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs1 6s 2r 362make r the subject in 2 2r 36 6sr 18 3s 16. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs1 6s 2r 362make r the subject in 2 2r 36 6s r 18 3s substitute into 1 17. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs1 6s 2r 362make r the subject in 2 2r 36 6s r 18 3s substitute into 1 A s 2 18 3s s 18. e.g. A rope 36 metres in length is cut into two pieces. The first is bentto form a square, the other forms a rectangle with one side thesame length as the square.Find the length of the square if the sum of the areas is to be amaximum. s rs s A s 2 rs1 6s 2r 362make r the subject in 2 2r 36 6s r 18 3s substitute into 1 A s 2 18 3s s A s 2 18s 3s 2A 18s 2s 2 19. A 18s 2s 2 20. A 18s 2s 2 dA 18 4s ds 21. A 18s 2s 2dA 18 4sds d2A2 4 ds 22. A 18s 2s 2dA 18 4sds d2A2 4 ds dAStationary points occur when0 ds 23. A 18s 2s 2dA 18 4sds d2A2 4 ds dAStationary points occur when0 dsi.e. 18 4s 0 9s 2 24. A 18s 2s 2dA 18 4sds d2A2 4 ds dAStationary points occur when0 dsi.e. 18 4s 0 9s 2 9 d2A when s , 2 4 0 2 ds 25. A 18s 2s 2dA 18 4sds d2A2 4 ds dAStationary points occur when0 dsi.e. 18 4s 0 9s 2 9 d2A when s , 2 4 0 2 ds1 when the side length of the square is 4 metres, the area is a maximum2 26. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume. 27. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume. rh 28. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h1h 29. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h112 r 2 2rh2h 30. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h112 r 2 2rh2 make h the subject in 2h 31. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h112 r 2 2rh2 make h the subject in 2h2rh 12 r 212 r 2 h2r 32. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h112 r 2 2rh2 make h the subject in 2h2rh 12 r 212 r 2 h2r substitute into 1 33. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h112 r 2 2rh2 make h the subject in 2h2rh 12 r 212 r 2 h2r substitute into 1 12 r 2 V r 2 2r 34. (ii) An open cylindrical bucket is to made so as to hold the largestpossible volume of liquid.If the total surface area of the bucket is 12 units 2, find the height,radius and volume.rV r 2 h112 r 2 2rh2 make h the subject in 2h2rh 12 r 212 r 2 h2r substitute into 1 12 r 2 V r 2 2r 1 V 6r r 3 2 35. 1 V 6r r 32 36. 1V 6r r 32 dV3 6 r 2 dr2 37. 1 V 6r r 3 2 dV 3 6 r 2dr2 d 2V2 3r dr 38. 1 V 6r r 3 Stationary points occur when dV0 2 dr dV 3 6 r 2dr2 d 2V2 3r dr 39. 1 V 6r r 3 Stationary points occur whendV 0 2dr dV 33 6 r 2 i.e. 6 r 2 0dr22 d 2V3 22 3r r 6 dr2r2 4 r 2 40. 1 V 6r r 3 Stationary points occur whendV 0 2dr dV 33 6 r 2 i.e. 6 r 2 0dr22 d 2V3 22 3r r 6 dr2r2 4 d 2Vr 2when r 2, 2 6 0 dr 41. 1 V 6r r 3Stationary points occur when dV0 2 dr dV 3 3 6 r 2i.e. 6 r 2 0dr2 2 d 2V 3 22 3rr 6 dr 2 r2 4 d 2V r 2when r 2, 2 6 0 dr when r 2, V is a maximum 42. 1 V 6r r 3Stationary points occur when dV0 2 dr dV 3 3 6 r 2i.e. 6 r 2 0dr2 2 d 2V 3 22 3rr 6 dr 2 r2 4 d 2V r 2when r 2, 2 6 0 dr when r 2, V is a maximum12 2 2when r 2, h 222 43. 1 V 6r r 3Stationary points occur when dV0 2 dr dV 3 3 6 r 2i.e. 6 r 2 0dr2 2 d 2V 3 22 3rr 6 dr 2 r2 4 d 2V r 2when r 2, 2 6 0 dr when r 2, V is a maximum12 2 2when r 2, h V 2 2 222 82 44. 1V 6r r 3Stationary points occur whendV 02 drdV 3 3 6 r 2i.e. 6 r 2 0 dr2 2d 2V 3 2 2 3rr 6dr 2r2 4d 2V r 2 when r 2, 2 6 0dr when r 2, V is a maximum 12 2 2 when r 2, h V 2 2 2 22 8 2 maximum volume is 8 units 3 when the radius and height are both 2 units 45. Exercise 10H; evens (not 30)Exercise 10I; evens