Sail Construction Objective Level Prerequisites Platform Instructor's Notes Credits

Objective: To demonstrate the strategy for constructing a triangular sail with maximum area.

Level:  This demo is available for three levels: prealgebra, algebra (with graphing) and calculus.

Prerequisites: Knowledge of the area of a triangle and appropriate background for the selected level of the demo.

Platform:  No computer system is necessary. Calculators can be used with each level as deemed appropriate by the instructor.

Instructor's Notes:  We present three procedures, prealgebra, algebra (possibly using
a calculator), and calculus to demonstrate to students in different courses a basic approach
to the common problem of maximizing a particular construction. The instructor can present
the introductory material that follows, and then use the course appropriate approach for
determining basic steps to this maximization problem. After the presentation students should
be able to apply the strategy to similar problems. The instructor can determine how much
student activity should support the basic demonstration steps. Having a pole available
for demonstration purposes has a strong appeal so that a visual stimulus can support the
discussion.

(A pole of actual length is not required. A piece of dowel wood several feet long makes a
nice model. In fact, tinker toys have been used successfully since you can really build the
triangular frame with them. If you were doing this as an introduction to a student activity, 
pieces of string can be used and cut by teams to build triangles and to compute the areas. 
The use of licorice whips in place of string, with the team that produces the triangle with 
largest area getting to keep the whips, introduces a bit of competition.)

Introduction: A pole 20 meters long is to be used to construct a triangular sail (frame). 

Figure 1.

The pole is to be cut into two pieces; a horizontal piece and a vertical piece. The two 
pieces are joined together to form a "T" shape with the "T" positioned at the midpoint 
of the horizontal piece. Figure 1 illustrates the construction. 

Figure 2.

Naturally to complete the sail construction we overlay this frame with a triangular piece of sail
material. Thus we get a triangle shown in Figure 2.

To take advantage of the wind to drive the sail boat we want to cut the 20 meter pole so
that the triangular sail has an area as large as possible. 

Figure 3.

Denote the length of the horizontal piece as x and the length of the vertical piece as y.

What is the area of the triangular sail?

area =_______________
                   (answer)
 

Prealgebra Approach to determining the sail with largest area.

Given that we choose to cut the 20 meter pole as follows, determine the corresponding
length of the vertical piece and the area of the triangle.

x = 3     y = __________ area = __________

x = 5     y = __________ area = __________

x= 12    y = __________ area = __________

One way to determine the lengths of the horizontal and vertical pieces that give a maximum
area is to construct a table of values.

Our table will consist of the length of the horizontal piece, denoted by x, the length of the 
vertical piece, denoted by y, and the corresponding area of the triangular sail.

To keep things simple we assume that we have only a meter stick to measure with, hence we
can cut the pole into lengths containing a whole number of meters. To facilitate the 
construction of the table fill in the blanks in the table which appears next. (Here the demo 
can use student involvement to aid in filling in the required data.)
 

x	y	Area		x	y	Area
1				11
2				12
3				13
4				14
5				15
6				16
7				17
8				18
9				19
10				20

From inspection of the table what are the dimensions of the triangle that will give the
triangular sail the largest area?

x = __________ y =___________

At this point the instructor can ask if we really have the sail of largest area since we restricted
the dimensions to be integers. If a student activity is planned to complement this demo, 
several calculations using a value of x around that determined above can be performed to 
provide further evidence. Of course, this points out the need for a careful approach which 
needs further mathematics.

Algebra Approach to determining the sail with largest area.

The area of the triangle in Figure 3 is A = 1/2 x y.

Rewrite this expression for the area in terms of the single unknown x. 
(Hint: Recall that the pole is 20m.)

A = ____________.

Plot the expression in your answer above for x between 0 and 20. (Remind students how to
do this by setting up table of values of (x, A), or if calculators with graphics are available, 
guide them through the procedure. If a table is used, caution them that they will need quite 
a few values of x for the table to get an accurate picture.)

What type of curve is generated? _________________

As accurately as possible estimate the value of x at which the graph is highest. 
(Restrict yourself to whole number values for x and recall that the dependent variable
is area.)

Maximum area occurs when x = __________.

What the values of x and y that give the triangle of largest area?

Calculus Approach to determining the sail with largest area.

The area of the triangle in Figure 3 is A = 1/2 x y.

Rewrite this expression for the area in terms of the single unknown x. (Hint: Recall that the
pole is 20m.)

A = ____________.

Using the expression you developed immediately above, apply the max-min strategy from
calculus to determine the value of x that makes the area of the triangle a maximum.
(At this point you can carefully go through the steps for setting up a max-min analysis.)

Maximum area occurs when x = __________.

What the values of x and y that give the triangle of largest area?

Dr.Roseanne Hofman
Department of Mathematics 
Montgomery County Community College

and is included in Demos with Positive Impact with her permission.