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Title
Pen for a Pony
Problem Statement
To make a pen for his new pony, Ted will use an existing fence as one side of
the pen. If he has 96 meters of fencing,
what are the dimensions of the largest pen he can make?
Problem setup
Given the perimeter of 3 sides
of a quadrilateral, the fourth side being a line of indefinite length, find the
quadrilateral with the largest area.
This is a traditional perimeter-area problem. Find the largest area with a given
perimeter. This problem is similar to
ones we work in class with pentominos dealing with perimeter and area.
Plans to
Solve/Investigate the Problem
To solve this perimeter-Area problem I will first try to visualize the
conditions and several pens that will satisfy them. I will also try to recall the information I
have previously learned about perimeter and area with relationship to my
knowledge of quadrilaterals. I will then
sketch a few different quads and give them dimensions. After calculating the perimeter and area of
my sketches I will try and make conjectures about the relationships between the
quads and there perimeter and area.
After I analyze the sketches and have some understanding about the
relationships I will construct a quadrilateral and calculate the perimeter and
area using Geometer’s Sketchpad. Once I
have made my constructions and calculations on GSP I will manipulate the quad
by dragging on its vertices as well as sides to form various types of
quads. As I make each different quad I
will pay close attention to their resulting perimeter and area. Finally, I will make conjecture about which
quad would represent the pony pen with the greatest area.
Investigation/Exploration
of the Problem
After visualizing the type of quads that satisfy the conditions for the
pen begin to sketch them. Start with
some line that represents the existing fence.
Then use the 96 meters to make different quads using the existing fence
as the forth side. Make one
quadrilateral wide with its long side running parallel to the existing fence so
that it is at least twice as long as the other sides. Draw one quadrilateral so that the longest
sides are perpendicular to the existing fence so that they are twice as long as
the shortest side. Finally, make one quadrilateral
that is a square, equal on all sides.
Once the sketches have been done assign them appropriate dimensions. After the figures have been constructed
calculate the area of each. Notice that the square has the biggest area. Using our constructions and inductive
reasoning we may generalize that for a fixed perimeter those that mark out a
square will have the greatest area.
Extensions of the Problem
Suppose
Ted needed his fence to be triangular for some important reason. What would the dimensions of the triangle
with the largest area be? What about a
pentagon?
Author & Contact
Nathan Little
nlittle@aug.edu
Link(s) to resources, references,
lesson plans, and/or other materials
My constructions
Intermath
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