- Visualisation of the Problem/Geometrical approach
The problem can easily be reduced to a two-dimensional problem, if you just look at the profile of the water gutter. Then the actual question is: for which angle does this quadrangle ( a trapezium - to be precise) have the biggest area? This trapezium can be constructed in the geometry part of the tool, so that the problem can be visualized dynamically.
By looking at the measured area you can conjecture that an angle of 60° leads to a maximal area.
The figure resulting for this angle is half of a regular hexagon.
A regular hexagon is the hexagon where the area is maximal when the perimeter is given.
- Table
Beside or instead of the geometrical construction you (one ?) can also collect the data of the area for different angles - in a systematical way which could also lead to an assumption/conjecture. - Graph
The graph shows the area as a function of the angle. It can be obtained in two ways: either directly by collecting the data for the angle and the area when moving the angle in the geometrical construction (in this case the table should be disabled) or by plotting the points that correspond to the table (as we normally do in the classroom). - Analytical Solution
The area as a function of the angle can be expressed as a trigonometric function. By calculating the coordinates of the vertex in the interval [0, 2π] we (one) get(s) the result.
Are all results equivalent? What are the differences and why? Which method is the best? Which is the fastest?