1. Optimization questions
MAP4C C2 – explain the significance of optimal dimensions in real-world applications, and determine optimal dimensions of 2-dimensional shapes and 3-dimensional figures.

2. q1
A hobby farmer is creating a fenced exercise yard for her horses. She has 900 m of flexible fencing and wishes to maximize the area. She is going to fence a rectangular or a circular area. Determine which figure encloses the greater area.
Rectangular case: the max area for this would be a square.
Using Perimeter :
900m = 4s
so s = 900/4
= 225 m
Area = 225 x 225
= m2

3. q1 Circular case: the circumference of a circle: C = 2πr 900 = 2 π r
Area of a circle:
A= πr2
= π(143.24)2
= m2
The circular pen provides the greatest area with a radius of 143 m.

4. q2
Among all rectangular prisms with a given surface area, a cube has the maximum volume.
Among all rectangular prisms with a given volume, a cube has the minimum surface area.
Yasmin is constructing a rectangular prism using exactly 96 cm2 of cardboard. The prism will have the greatest possible volume. Determine its dimensions and volume.
Solution: A cube has the maximum volume. There are 6 equal squares that make up the cube:
SA = 6s2 = 96
s = 4 (So a 4cm x 4cm x 4cm cube)
V = s3 = 43 = 64 cm3

5. s = 9 inches (A 9” x 9” x 9” cube)q3
Matthew is constructing a rectangular prism with a volume of 729 cubic inches. The prism will have the least possible surface area. Determine its dimensions and surface area.
Solution: A cube has the minimum surface area. There are 6 equal squares that make up the cube:
V = s3
729 = s3
s = 9 inches (A 9” x 9” x 9” cube)
SA = 6s2
= 6(9)2
= 486 square inches

6. When h = 2r, the minimum surface area is achieved (cylinders).q4
Naveed is designing a can with volume 350 mL. What is the minimum surface area of the can? Determine the dimensions of a can with the minimum surface area.
As 1 mL = 1 cm3, the volume is 350 cm3.
V = πr2h
When h = 2r, the minimum surface area is achieved (cylinders).
350 = πr2(2r)
= 2πr3
r3 = 350/2π
r = 3.8 cm
h = 2(3.8) = 7.6 cm

7. q4
Naveed is designing a can with volume 350 mL. What is the minimum surface area of the can? Determine the dimensions of a can with the minimum surface area.
SA = 6πr2 = 6 π(3.8)2
= cm2