Вашата кошница е празна
A practical application of the functions
We are looking for the answer to the following problem. We are sailing on a lake (or on a sea) with a northerly wind, and our aim is still to sail north as quickly as possible under the circumstances. How shall we manoeuvre, or cruise to use the correct terminology, to achieve this?
We can change the following angles:
\latex{\alpha}: the angle included between the direction of the wind and the direction of the sail,
\latex{\beta}: the angle included between the axis of the hull and the direction of the sail, and
\latex{\gamma}: the angle included between the axis of the boat and the direction of east (when heading north).
We know that \latex{\alpha+\beta+\gamma=90°} (Figure 37).
We are going to show that even the wind blowing from north has a component that makes the boat move towards north if the sail and the axis of the boat are in the correct direction.
\latex{E}
\latex{\beta}
\latex{\gamma}
sail direction
axis of the boat
\latex{\alpha}
\latex{N}
Figure 37
\latex{P_{axis}=\left|P_\bot \right|\times \sin \beta }
\latex{P_\bot =\left|\vec{p} \right|\times \sin \alpha }
\latex{\beta}
\latex{E}
\latex{\vec{p}}
\latex{\alpha}
\latex{N}
Figure 38
wind direction
sail direction
axis of the boat
\latex{P_{north}=\left|P_a\right|\times\sin \gamma }
\latex{P_a}
\latex{E}
\latex{\gamma}
Figure 39
Let us denote the force of north wind blowing by \latex{\vec{p} }. The thrust of this force applied to the sail is a force perpendicular to the plane of the sail (Figure 38):
\latex{P_\bot =\left|\vec{p} \right|\times \sin \alpha }.
The component of the force applied to the sail in the direction of the axis of the boat makes the boat move forward. As the angle included between the force and the axis of the boat is \latex{\beta}:
\latex{P_{axis}=\left|P_\bot \right| \times \sin \beta }.
The direction of the axis of the boat, i.e. the heading direction of the boat and the direction of east include an angle of \latex{\gamma}. The northerly component of the force pointing to the heading direction defines the magnitude of the force that makes the boat move towards north (Figure 39):
\latex{P_{north}=\left|P_{axis} \right| \times \sin \gamma }.
To summarise: The northerly component of the wind blowing from north with a force of \latex{\vec{p} } is:
\latex{P_{north}=\left|\vec{p} \right| \times \sin \alpha \times \sin \beta \times \sin \gamma }.
So we are looking for the maximum of the following expression
\latex{\left|\vec{p} \right| \times \sin \alpha \times \sin \beta \times\sin \gamma },
where \latex{\alpha +\beta +\gamma =90°}, and \latex{\alpha, \beta, \gamma} are acute angles.
As \latex{\sin \alpha ,\;\sin \beta, \;\sin \gamma } are positive numbers, we can apply the inequality of arithmetic and geometric means:
\latex{\sin \alpha \times \sin \beta\times \sin \gamma \leq \left\lgroup\frac{\sin \alpha +\sin \beta +\sin \gamma }{3} \right\rgroup^3 }
and the equality holds only if \latex{\sin \alpha = \sin \beta= \sin \gamma}, which is satisfied only in the case of \latex{ \alpha = \beta= \gamma} because of the conditions.
Let us also use the result of exercise \latex{ 4 } of the chapter about the sine function from the book for year \latex{ 10 }:
\latex{\frac{\sin \alpha +\sin \beta +\sin \gamma }{3} \leq \sin \frac{\alpha +\beta +\gamma }{3}=\sin 30°=\frac{1}{2} },
as \latex{\alpha +\beta +\gamma =90°}. (The idea in the solution of the exercise mentioned is that the sine function is concave on the interval \latex{\left[0;\pi\right]}.) The equality holds in the case of \latex{ \alpha = \beta= \gamma=30°} here too.
It is obtained that the largest value of the product with three factors is \latex{\frac{1}{8}}:
\latex{\sin \alpha \times\sin \beta \times \sin \gamma\leq\frac{1}{8}}.
It means that the boat can cruise towards north with \latex{\frac{1}{8}} of the speed that could be reached heading towards south. The angle included between the axis of the boat and the direction of east should be \latex{ 30º }, and the angle included between the sail and the axis of the hull should also be \latex{ 30º }.
