MPM1D1
Optimizing Measurements
Name _________________
Optimizing means finding values that make a given quantity (such as area, surface area or volume)
the GREATEST possible (or the LEAST possible) given certain conditions.
When building an enclosure, it is economical to build the largest enclosure possible with the least
amount of fencing, for instance, to keep costs low.
PART A
1. What is the largest enclosure area one can completely surround with ___ m of fence?
(ie, Find the rectangle with a perimeter of ____ m which has the OPTIMUM AREA.)
Rectangle No.
Width (m)
Length (m)
Area (m2)
Perimeter (m)
1
2
3
4
5
6
7
8
Construct a scatter plot of area vs. width.
Area vs. Width
26
24
22
20
Area (m2)
18
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
9
10
11
MPM1D1
PRACTICE: Optimizing Measurements
Name ___________________
PART B
Find a group who have different measurements.
What is the perimeter for their shape?
What are the dimensions?
What is the largest enclosure area one can completely surround?
Conclusion:
What do you notice about the graph?
Optimum area of a rectangle is one where length ________ width (ie a __________ )
If only 3 sides are enclosed with 24 m of fencing, you would achieve OPTIMUM AREA by building a
________________ enclosure.
OPTIMIZING 3D SHAPES
In manufacturing and packaging, it is often desirable to achieve maximum volume with minimum
surface area. This maximizes the amount a container will hold while minimizing the cost of packaging.
PART A
1. Each of the square-based prisms has a surface area of 24 cm2. Calculate the area of the base and
the volume of each prism. Record your data in a table.
0.5 cm
5.5 cm
2 cm
3 cm
2 cm
1 cm
3 cm
2 cm
1 cm
Prism No.
1
2
3
Side length of
Base (cm)
Area of Base
(cm2)
Surface Area
(cm2)
24
24
24
Volume
(cm3)
2. Which square-based prism has the maximum volume? Describe the shape of this prism compared
to the others.
3. Each of the square-based prisms has a surface area of 54 cm2.
a) Calculate the area of the base and the volume of each prism. Record your data in a table.
0.2 cm
13 cm
3 cm
5 cm
3 cm
5 cm
3 cm
1 cm
1 cm
b) The Surface Area of a square-based prism is found with the formula: SA = 6s2.
Predict the dimensions of the square-based prism with maximum volume if the surface area is
(i) 150 cm2.
(ii) 6000 cm2
Conclusion: What conclusion can you make about the maximum volume of a square-based prism
with a given surface area?
PART B
1. Each of the cylinders has a surface area of 375 cm 2. Calculate the area of the base, the height and
the volume of each prism. Record your data in a table.
3.75 cm
22 cm
10 cm
r = 7 cm
r = 5 cm
r = 3 cm
Cylinder No.
1
2
3
Radius (cm)
Height (cm)
Surface Area
(cm2)
Volume
(cm3)
375
375
375
2. Which cylinder has the maximum volume? Describe the shape of this cylinder compared to the
others.
Conclusion: What conclusion can you make about the maximum volume of a cylinder with a given
surface area?
MPM1D1
PRACTICE: Optimizing Measurements
SA (square based pyramid) = 6s2
Name ____________________
V(sbp) = s3
1. A box of powdered laundry detergent is packaged in a square-based prism box. What is the smallest amount
of cardboard that could be used to make a box that holds 4000 cm3 of detergent?
2. What are the dimensions of a square-based prism box that provides the maximum volume using 2500 cm3
of cardboard?
3. A cylindrical can must hold 376 mL of juice. Determine the dimensions of the can that require the least
amount of aluminum.
4. A container is needed that has a fixed volume of 1000 cm3. Which three-dimensional figure will use the least
material in its construction: a cube, a cylinder whose h = d, or, a sphere?