To find a maximum or minimum

Example A piece of wire 20 cm long is bent into the shape of a rectangle. Find the maximum area it can enclose.
Let the length be x, then the width will be 10 – x A = x(10 – x) = 10x – x2 = 10 – 2x = 0 gives x = 5 = – 2 implying a maximum. The area is maximum when each side is 5 cm. The maximum area is 25cm2	(long side + short side = half of 20) (Note the area is a maximum when the shape is a square.)
Example The velocity of a car, v ms-1 between two road junctions is modelled by v = 3t – 0.2t2 for 0  t  15 where t is the time in seconds after it sets off from the first junction. Find the maximum speed.
For maximum speed = 3 – 0.4t = 0 0.4t = 3 t = = 7.5 (seconds) = – 0.4 implying a maximum. Maximum speed = 3t – 0.2t2 = 3  7.5 – 0.2  7.52 = 11.25 ms-1	Note = acceleration When the speed reaches a maximum, the acceleration is zero.
Example The function p = x3 – 18x2 + 105x – 88 models the way the profit per item made, p pence, depends on x, the number produced in thousands. Find the maximum and minimum values of p and sketch a graph of p against x.
For max/min: = 0 Simplify the equation: = 0 then solve it = 0 x = 5 or 7 = 6x – 36 is negative when x = 5 (maximum) and positive when x = 7 (minimum) When x = 5, p = 53 – 18  52 + 105  5 – 88 = 112 When x = 7, p = 73 – 18  72 + 105  7 – 88 = 108 There is a maximum point at (5, 112) and a minimum point at (7, 108).
The model predicts a peak on the graph when 5000 are produced, the profit per item then being £1.12. The profit per item falls to £1.08 when 7000 are produced before rising again.
Example A cylindrical hot water tank is to have a capacity of 4 m3. Find the radius and height that would have the least surface area.
Substitute for h to find a formula for the surface area in terms of just one variable, r: Simplify: Differentiate: = 0 for max/min Solve the equation: is positive This implies minimum surface area. Also = 1.72... The tank with minimum area has radius 0.86 m and height 1.72 m (to 2 dp).	Formulae for a cylinder: Surface Area Volume = 4 gives The minimum surface area can be found from the formula above: The minimum surface area is 13.9 m2 (to 1 dp)

a) Show that the area of the enclosure is given by:

c) Calculate the maximum possible area.

6 A farmer has 100 metres of fencing to use to
make a rectangular enclosure for sheep as shown.
He will use an existing wall for one side of the enclosure
and leave an opening of 2 metres for a gate.

a) Show that the area of the enclosure is given by:

7 A farmer has 100 metres of fencing to use to

a) Show that the area of the enclosure is given by:

a) Show that if the original rectangle of card

b) Find the value of x that will give the maximum possible volume.

9 Repeat question 8 starting with

10 A closed tank is to have a square base and capacity 400 cm³

b) Find the value of x that will give the minimum surface area.

11 Find the minimum surface area of an open-topped tank with a square base and capacity

12 A soft drinks manufacturer wants to design a cylindrical can

13 Repeat question 20 for a can to hold 1 litre of drink.

14 An aircraft window consists of a central rectangle and two

A window is required to have an area of 1 m²

Teacher Notes

• 
  solving maximum and minimum problems
  

Students need to be able to:

• 
  differentiate polynomials;
  
• 
  solve linear and quadratic equations;
  
• 
  sketch curves.
  

It is recommended that you use the Nuffield activity ‘Stationary Points’ before this activity.

1 8 ms^-1

2 20 m

3 40 m

5 a) 25.5 m b) 650.25 m²

6 a) 25.5 m b) 1300.5 m²

7 a) 25.5 m b) 2601 m²

8 b) 10 cm c) 18 000 cm³

9 a) x = 20 giving 144 000 cm³ b) x = 7.85 giving 8 450 cm³ (3 sf)

10 b) 7.37 cm c) 326 cm² (3 sf)

11 x = 9.28 giving 259 cm² (3 sf)

12 r = 4.30 giving 349 cm² (3 sf)

13 r = 5.42 giving 554 cm² (3 sf)