8.8: Optimum Volume and Surface Area

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8.8 Optimum Volume and Surface Area

• MPM1D1
• March 2008
• J. Pulickeel

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What Is Optimal Area and Volume?
When we are talking about optimal area and volume
we want to MAXIMIZE the VOLUME and MINIMIZE the
AREA
V 1000 u3
V 1000 u3
V 1000 u3
V 1000 u3
SA 600 u2
SA 739.2 u2
SA 850 u2
SA 669.2 u2
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Which shape is the most Optimal?

• A SPHERE has the largest volume and smallest
  surface area.
• A 3D shape that is CLOSEST to the shape of
  sphere will have the next largest volume
• A CUBE is the rectangular prism with the largest
  volume

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Which shape would have the optimal Volume if the
Surface Area is the same?
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2
3
5
How could I increase the volume of these shapes
without changing the surface area?
2
3
1
The height and diameter should be the same
Change this cylinder into a cylinder that is
closer to a sphere/cube
Change this rectangular prism into a cube
Change this oval into a sphere
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Find the maximum volume of a cube with a surface
area of 1200cm2

• SACUBE 6l2
• SACUBE 1200cm2

VCUBE l3
1200cm2 6l2
VCUBE (14.14cm)3
1200cm2 6l2 6 6
VCUBE 2828.4cm3
200cm2 l2
14.14cm l
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Find the maximum volume of a cylinder with a
surface area of 1200cm2

• We need a cylinder where the height is equal to
  the diameter, and the SA must equal 1200cm2

Height (cm) Diameter(cm) Equal to height Radius (cm) ½ the height SA 2pr2 h(2pr) This has to equal 1200cm2 Volume (cm3) V hpr2