1. 8.8: Optimum Volume and Surface Area
MPM1D1
March 2008
J. Pulickeel

2. 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 5 40 6
27.8 7
20.4 10
V = 1000 u3
V = 1000 u3
V = 1000 u3
V = 1000 u3
SA = 600 u2
SA = u2
SA = 850 u2
SA = u2

3. 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

4. Which shape would have the optimal Volume if the Surface Area is the same?1 3 2

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
D = 2r=h h d h l l w h r

6. 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 = cm3
200cm2 = l2
14.14cm = l

7. 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