Minimize the Surface Area of a Square‐Based Prism
Minimizing surface area for a given volume is important when designing packages and containers to save on materials and reduce heat loss.
Example 1: The Acme Box Company designs and manufactures boxes, cans, and other containers. It has been hired to design a box to hold 64 cm^3 of a certain product. The box could have different shapes. What dimensions require the minimum amount of cardboard? What shape is this?
– The dimensions of the box that require the minimum amount of cardboard to hold 64 cm3 of a certain product are 4 cm by 4 cm by 4 cm.
– The shape of the box is a cube.
Conclusion: For a square-based prism with a given volume, the minimum surface area occurs when the prism is a cube.
Volume:
V=s s s
=s3
Surface Area:
S.A.=6s2
S is the side length of the cube.
Example 2: a) Determine the dimensions of the squarebased prism box with a volume of 750 cm3 that requires the least material to make. Round the dimensions to the nearest tenth of a centimetre.
Solution: Given: V=750 cm3
Find s=?
Vcube=s3
750=s3
S=3750 or s=75013
S=9.1 cm
The dimensions of the square-based prism are 9.1 cm by 9.1 cm by 9.1 cm.
b) Find the amount of cardboard required to make this box
S.A.cube=6s2
=69.12
=6(82.81)
=496.86
The amount of cardboard required to make this box is 496.86 cm2.
Example 3: Talia has been asked to design an insulated squarebased prism container to transport hot food. When hot food is placed in the container, it loses heat through the container’s sides, top, and bottom. To keep heat loss to a minimum, the total surface area must be minimized.
a) Find the interior dimensions of the container with volume 145 000 cm3that has minimum heat loss
V=145000 cm3
V=s3
145000=s3
s3=145000
s=3145000 or s=14500013
s=52.54
The interior dimensions of the container with volume 145000 cm3 that has minimum heat loss are 52.54 cm by 52.54 by 52.54 cm.
b) What other factors might Talia consider?
-The product the box is designed for
-Safety
-Eye appealing.
