8.4 Investigating te Maximum Area of a Rectangle
May 28, 2014
8.1 Investigating the Maximum Area of a Rectangle
8.4 Investigating the Maximum Area of a Rectangle Learning Goals
determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles and by examining various values of the side lengths and the perimeter as the perimeter stays constant.
8.1 Investigating the Maximum Area of a Rectangle
8.4
Investigating the Maximum Area of a Rectangle
Sometimes you need to determine the maximum area maximum area of a shape. When the perimeter of a rectangle is fixed, what dimensions create the optimum shape?
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8.4 Investigating te Maximum Area of a Rectangle
May 28, 2014
8.1 Investigating the Maximum Area of a Rectangle
Action
a)
Create as many different rectangles as you can that have a perimeter of 16 cm. Record the dimensions and the area in the table.
Width
(units)
Length
(units)
Area
(units2)
8.1 Investigating the Maximum Area of a Rectangle
b) Represent the pattern in the width and area of the rectangles using a table of values and a graph.
16 cm
Fixed perimeter: __________
Area vs. Width
20
Width of Area of
rectangle (cm) rectangle (cm2 )
Area of rectangle (cm2)
B
C
15
D
E
E
10
F
5
0
2
4
6
Create a scatte
information in y
Draw a circle on
Draw a line or c
where you find t
appropriate.
What is the corr
Write the dimen
with the maximu
What can you co
dimensions that
In general, do y
area for your fix
shape for a rect
What shape is th
perimeter is alw
Why or why not
8
Width of rectangle (cm)
2
8.4 Investigating te Maximum Area of a Rectangle
May 28, 2014
8.1 Investigating the Maximum Area of a Rectangle
When the perimeter must stay the same, what do you notice about the rectangle with the largest area?
8.1 Investigating the Maximum Area of a Rectangle
Summary
same perimeter can • Different rectangles with the Reveal
enclose different amounts of area.
• For any rectangle with a fixed perimeter, there is Reveal
between the width or a non­linear relationship
length of the rectangle and its area.
• For a rectangle with a fixed perimeter, the greatest or maximum area occurs when the Reveal
length and the width are equal
. The maximum area occurs when the rectangle is a square with either whole­number or decimal dimensions.
• In real­world situations involving a rectangle with a fixed perimeter, the dimensions of the rectangle may be restricted to whole numbers. The greatest Reveal
or maximum area will occur when the dimensions of the rectangle are amounts that are either equal or close to each other.
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8.4 Investigating te Maximum Area of a Rectangle
May 28, 2014
8.1 Investigating the Maximum Area of a Rectangle
Knowing this and using algebra, we can determine the dimensions and area of the rectangle with the largest area quickly.
8.1 Investigating the Maximum Area of a Rectangle
Determining Optimum Area ­ Three Sides
What is the largest rectangular area that can be built with a 24 m fence on
the side of a building?
width
width
length
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8.4 Investigating te Maximum Area of a Rectangle
May 28, 2014
8.1 Investigating the Maximum Area of a Rectangle
Summary ­ Optimum Area ­ Three Sides
width
width
length
A rectangle with a border on three sides has a maximum area and a minimum perimeter when the side without a border and its opposite side are twice the length of the other two sides.
length = 2 x width
8.1 Investigating the Maximum Area of a Rectangle
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Attachments
8.1 Maximizing Area for a Fixed Perimeter.gsp
am9_8_1_tn_guided.doc
am9_8_1_tn_open.doc
am9_ blm_8_1a_minds_on.doc
am9_ blm_8_1b_action.doc
am9_ blm_8_1c_action_guided.doc