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