Free-Standing Mathematics Activity Maximum and minimum problems

1
Free-Standing Mathematics ActivityMaximum and
minimum problems
2
Manufacturers use containers of different shapes
and sizes.
How can manufacturers design containers to

• hold as much as possible
• use as little material as possible?

In this activity you will use graphs to solve
such problems.
3
A drinks can must hold 330ml
The manufacturer wants to find the dimensions
with the minimum surface area.
330 pr2h
V pr2h
S 2pr2 2prh
Think about Which formulae do you think will be
needed to solve this problem?
S 2pr2 2pr
Think about How can a minimum value for S be
found?
To find the minimum area, draw a graph of S
against r on a spreadsheet or graphic calculator.
4
Think about How can a more accurate minimum be
found?
Think about What is the minimum surface area?
5
Using smaller increments of r near the minimum
Minimum S
h 7.490 cm
Check this gives a volume of 330 cm3
Minimum surface area is 264.36 cm2 when r
3.745 cm and h 7.490 cm
6

• Reflect on your work
  
• Give a brief outline of the method used to find
  the minimum surface area for a can holding 330 ml
  of drink.
• What difference would it make to the surface
  area if a cuboid with square cross-section was
  used for holding the drink?
• Do you think a cylinder is the best shape to use?
  Why?
• Can you find any connections between the types of
  equation leading to a maximising problem, and
  those which lead to a minimising problem?