© Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Galileo’s projectile model.
© Nuffield Foundation 2012 Free-Standing Mathematics Activity Maximum and minimum problems.
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Transcript of © Nuffield Foundation 2012 Free-Standing Mathematics Activity Maximum and minimum problems.
© Nuffield Foundation 2012
– hold as much as possible– use as little material as possible?
Manufacturers use containers of different shapes and sizes.
In this activity you will use graphs to solve such problems.
How can manufacturers design containers to:
© Nuffield Foundation 2012
A drinks can must hold 330ml The manufacturer wants to find the dimensions with the minimum surface area.
Capacity330 cm3
radiusr cm
heighth cm
V = r2h
S = 2r2 + 2rh
S = 2r2 + 2πr × 2330r
S = 2r2 + r660
2330rh =
330 = r2h
Think about …Which formulae do you think will be needed to solve this problem?
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.
© Nuffield Foundation 2012
0
100
200
300
400
500
600
700
0 1 2 3 4 5 6 7 8 9 10
Surf
ace
area
(cm
2 )
Radius (cm)
Surface area of a canS = 2r2 + r
660
Minimum area 260 cm2
when r = 3.7 cm
Think about…What is the minimum surface area?
Think about…How can a more accurate minimum be found?
© Nuffield Foundation 2012
264.3
264.4
264.5
264.6
264.7
3.60 3.65 3.70 3.75 3.80
Surf
ace
area
(cm
2 )
Radius (cm)
Surface area of a canS = 2r2 + r
660
h =
2330r
h = 7.490 cm
Check this gives a volume of 330 cm3
Minimum S
Minimum surface area is 264.36 cm2 when r = 3.745 cm and h = 7.490 cm
= 264.36 cm2
when r = 3.745 cm
Using smaller increments of r near the minimum
= 27453π330
© Nuffield Foundation 2012
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?