Maximum/Minimum

1. Maximum/Minimum NCEA Problems

2. The entrance to the Italian Gardens in Hamilton has a curved trellis on top of two vertical side-sections of equal height. The curved section can be modelled by the parabola where x is the horizontal distance in metres from the left side of the trellis and y is the height in metres of the curved parabolic section of the trellis above the top of the vertical sides. Use calculus to find the maximum height of the curved parabolic section of the trellis above the vertical sides.

3. To find where the maximum occurs, gradient is zero.

4. Substitute to find the maximum:

5. 2008 • A piece of wire of length 40 cm is cut into two pieces. One piece is bent to form a square, while the other is bent to form an equilateral triangle. Find the length of the wire used for the square if the sum of the areas of the two shapes is a minimum.(You may assume that the value found gives the minimum area.)

6. Area = • Perimeter = 4x + 3y = 40 60 x y

7. Area = • Perimeter = 4x + 3y = 40 60 x y

8. Area = 60 x y

9. Answer the question:Wire length is 17.4 cm 60 x y

10. 2007

11. The daily profit (in thousands of dollars) from sales of a chemical is given byP(x) = x3- 12x2 + 36x + 3, for 1 < x < 8where x is the number of tonnes of chemical sold daily.What is the minimum daily profit?

12. P(x) = x3- 12x2 + 36x + 3

13. P(x) = x3- 12x2 + 36x + 3 • Test for Minimum

14. P(x) = x3- 12x2 + 36x + 3 • Minimum daily profit is • Put this in context I.e. $3,000