1.6: Optimization I
Maximizing Area of Rectangles
MPM1D1: Grade 9 Academic Mathematics
Unit 1: Measurement Relationships and Optimization
Problem
The Cepsi~Cola Concert Series
Cepsi~Cola is holding a fundraiser concert to for the homeless in our area. The company was able to
sign Eminem for an outdoor concert outside Cardinal Ambrozic C.S.S. To ensure he has a secure playing
area, a rectangular fence is to be constructed around a field so that one side of the field is bounded by
the wall of the school. Determine the maximum area that can be enclosed if the total length of fencing
to be used is 500 m.
Modelling Through Diagrams
1.
How many fields are possible? Draw 3 fields and calculate their areas.
Perimeter =
b)
a)
SCHOOL WALL
Length of field (m)
)
c)
SCHOOL WALL
Width of field (m)
SCHOOL WALL
Area of field (m2)
2.
Do you think it is possible to enclose a larger area than these with 500 m of fencing? Explain
your answer.
3.
How can we calculate the length if we are given the width?
Modelling Through Tables
4.
Investigate the possible fields listed below. Enter your data into lists in a graphing calculator as
follows. Calculate the widths for each length knowing that: Perimeter = L + 2W
Width
Length
Area (A)
0
25
50
75
100
125
150
175
200
225
250
5.
What are the dimensions of the field with the greatest area? Is this the largest possible area?
Modelling Through Graphing
6.
What are the dimensions of the field with the greatest area? Is this the largest possible area?
7.
What is the relationship between length and width that will provide the largest area for a threesided rectangle?
Modelling Through Technology
1. Turn on the calculator and press the home
c
button ( )
2. Select option 1:New Document
3. Once prompted to save the document select
NO
4. Select option 4:Add Lists & Spreadsheet
5. Bring the cursor to the top of column A and
type the name of the independent variable
(width). Do the same for column B (length) and
C (area)
6. Enter the value of the width in the width
column.
7. Let the calculator calculate the length by
manipulating the formula P=2w + l to:
L=500 – 2W
Type formula as shown under the cell of
B:length
500 – 2width
Press ENTER
500 – 2width
8. Create a formula to calculate the area of the
rectangle by multiplying the length and width.
9. Now you are ready to graph. Press the home
c
button ( ). Select Data and Statistics (the
histogram icon
)
10. Bring the cursor to the right on the y-axis.
Select area by pressing enter.
11. Bring the cursor to the bottom on the x-axis.
Select width by pressing enter.
12. Estimate the dimension that gives the
maximum area.
Additional Problems
8.
On your own, determine the maximum area and dimensions of a similar field given the following
total amounts of fencing.
(i)
300 m
(ii)
800 m
(iii)
1500 m
(iv)
3000 m
Each solution should include a diagram, a chart, a graph, an explanation of the method used and
any calculations, and conclusions about the area and dimensions.
Homework (pg. 487)
9.2 #2 – 6