8.8: Optimum Volume and Surface Area
MPM1D1
March 2008
J. Pulickeel
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
10
10
10
6
27.8
6
55
40
7
20.4
7
V = 1000 u3 V = 1000 u3V = 1000 u3 V = 1000 u3
SA = 850 u2 SA = 739.2 u2SA = 669.2 u2
SA = 600 u2
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
Which shape would have the optimal Volume if the Surface Area is the same?
123
How could I increase the volume of these shapes without changing the surface area?
1 2 3Change this oval into a
sphere
Change this cylinder into a cylinder that is closer to a
sphere/cube
Change this rectangular prism into a cube
r
l
w
h
l
l
l
d
h
D = 2r=h
h
The height and diameter should be the
same
Find the maximum volume of a cube with a surface area of 1200cm2
SACUBE = 6l2
SACUBE = 1200cm2 VCUBE = l3
VCUBE = (14.14cm)3
VCUBE = 2828.4cm3
1200cm2 = 6l2
200cm2 = l2
1200cm2 = 6l2
6 6
14.14cm = l
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 = 2πr2 + h(2πr)This has to equal 1200cm2
Volume (cm3)V = hπr2
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