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Maximum and minimum problems

Maximum and minimum problems. Calculus AS 3.6. A rectangular block is constructed so that its length is twice its breadth. Find the least possible surface area of the block if its volume is 72. Step 1: Draw a diagram and label it. “ its length is twice its breadth ”. h. x. 2 x.

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Maximum and minimum problems

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  1. Maximum and minimum problems Calculus AS 3.6

  2. A rectangular block is constructed so that its length is twice its breadth. Find the least possible surface area of the block if its volume is 72

  3. Step 1: Draw a diagram and label it • “its length is twice its breadth” h x 2x

  4. Step 2: Write the equation for the relationship given • its volume is 72 h x 2x

  5. Step 3: Write the equation for what you are trying to maximise or minimise • “Find the least possible surface area” h x 2x

  6. Step 4: Use the information to write this equation in one variable only • “Find the least possible surface area”

  7. Step 5: Differentiate = 0 • “Find the least possible surface area”

  8. Step 6: Answer the question • “Find the least possible surface area”

  9. IF the size of a parcel sent through the post is limited by the fact that the sum of its length and girth must not exceed 2 m, find the volume of the largest rectangular parcel with a square base which may be posted.

  10. 1. Draw a diagram y x x

  11. 2. Write the equation for the relationship given y x x

  12. 3. Write the equation for what you are trying to maximise or minimise find the volume of the largest rectangular parcel y x x

  13. 4: Use the information to write this equation in one variable only find the volume of the largest rectangular parcel y x x

  14. 5: Differentiate = 0 find the volume of the largest rectangular parcel

  15. 6: Answer the question find the volume of the largest rectangular parcel Don’t forget units

  16. A feeding trough is made from three pieces of metal welded together, one rectangle of size 3k cm by 5k cm and two trapeziums. The length of the trough is 5k cm and its cross-section is shown in the diagram. (k is constant)

  17. Find the maximum volume of the trough. k k k (5k long)

  18. Diagram is given and we only need to use 1 variable, θ k k k (5k long)

  19. k k k (5k long)

  20. k k k k (5k long)

  21. k k k k (5k long)

  22. k k k k (5k long)

  23. k k k k (5k long)

  24. k k k k (5k long)

  25. k k k k

  26. Find the maximum volume of the trough. k k k k

  27. A circle of radius l is used to form a cone. Find the angle of the sector that forms a cone of the maximum volume.

  28. Think about how the cone is created l l θ l

  29. The arc length is the same as the circumference of the rim of the cone. r l l θ l

  30. Using Pythagoras’ theorem r l h l θ l

  31. Volume of the cone = r l h l θ l

  32. Volume of the cone =

  33. Substituting

  34. Clean it up

  35. Differentiate using product rule

  36. Differentiate using product rule Would give a minimum

  37. Differentiate using product rule

  38. If total revenue for a firm is given bywhere x is the number of units sold, find the number of units that must be sold to maximize revenue.

  39. The number of units that must be sold to maximize revenue is 40.

  40. A travel agency will plan a group tour for groups of 25 or larger. If the group contains exactly 25 people, the charge is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency?

  41. A travel agency will plan a group tour for groups of 25 or larger. If the group contains exactly 25 people, the charge is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency?R = number of people x cost

  42. A travel agency will plan a group tour for groups of 25 or larger. If the group contains exactly 25 people, the charge is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency?R = number of people x cost

  43. A travel agency will plan a group tour for groups of 25 or larger. If the group contains exactly 25 people, the charge is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency?R = number of people x cost Group size 22 or 23 for revenue $7560

  44. Suppose the production capacity for a certain commodity cannot exceed 30. If the total profit function for this commodity iswhere x is the number of units sold, find the number of items that will maximize profit.

  45. Suppose the production capacity for a certain commodity cannot exceed 30. If the total profit function for this commodity iswhere x is the number of units sold, find the number of items that will maximize profit.

  46. Suppose the production capacity for a certain commodity cannot exceed 30. If the total profit function for this commodity iswhere x is the number of units sold, find the number of items that will maximize profit.Check end point x = 30, P = $26,800

  47. End point x = 30, P = $26,800 gives the maximum

  48. A man at a point A on the shore of a circular lake with radius 2 km wants to be at point C diametrically opposite A on the other side of the lake in the shortest possible time. The ratio of his walking speed to his rowing speed is 2:1. At what angle to the diameter should he row?

  49. Check end points

  50. Time to row across A C 4 km

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