Water gutter
A water gutter with optimal capacity shall be constructed from three boards
with width 0.5 m each (see illustration). Show: The area of the cross section
is given by the function A with
A(\alpha) = 0,25 \times (1 + \cos(\alpha)) \times \sin(\alpha).
Find the angle \alpha so that the water gutter
has maximal capacity.
The idea
A geometrical problem is described and solved algebraically by using basic
geometrical formulae, functional knowledge or commands about how to find
an optimal value. The technology enables a dynamic visualization of the
situation to lead to a deeper understanding and an idea of solution to the
optimization problem on a low level - so that you only need dynamic visualization
to get an idea of the situation and a possible solution.
Aims
- The pupils should understand that depending on the variation of
the angle the area of the cross section changes. For this they investigate
a dynamic geometrical visualization and put up first conjectures for
min or max.
- The pupils use a given function which describes the situation to
make further investigations with in algebraic means.
- The pupils solve the problem by using the CAS technology (e.g. min
or max-command)
- The pupils confirm that the given function is indeed a correct description
of the situation
Detailed explanation of the aims ...
Prerequisites
- age of pupils: Upper secondary level, at least
16 years old
- previous knowledge:
- basic technical features of the CAS-device
- trig functions (esp. sine)
- min and max
- measuring the area of a trapezoid
CLASSROOM ACTIVITIES
letīs solve the task