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 pole of actual length is not required. A piece of dowel wood several
feet long makes a
Introduction: A pole 20 meters long is to be used to construct a triangular sail (frame).
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.
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
What is the area of the triangular sail? area =_______________
Prealgebra Approach to determining the sail with largest area. Given that we choose to cut the 20 meter pole as follows, determine
the corresponding
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
Our table will consist of the length of the horizontal piece, denoted
by x, the length of the
To keep things simple we assume that we have only a meter stick to measure
with, hence we
From inspection of the table what are the dimensions of the triangle
that will give the
x = __________ y =___________ At this point the instructor can ask if we really have the sail of largest
area since we restricted
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.
A = ____________. Plot the expression in your answer above for x between 0 and
20. (Remind students how to
What type of curve is generated? _________________ As accurately as possible estimate the value of x at which the
graph is highest.
Maximum area occurs when x = __________. What the values of x and y that give the triangle of largest area? x = _____ y = _____ 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
A = ____________. Using the expression you developed immediately above, apply the maxmin
strategy from
Maximum area occurs when x = __________. What the values of x and y that give the triangle of largest area? x = _____ y = _____ Area = _____
Credits: This demo was submitted by Dr.Roseanne Hofman
and is included in Demos with Positive Impact with her permission. 

Since 3/1/2002