Triangles

9000150501

Level: 
C
A man of height \(180\, \mathrm{cm}\) casts a \(200\, \mathrm{cm}\) shadow. At the same moment, a tree of an unknown height casts a \(35\, \mathrm{m}\) shadow. Find the height of the tree.
\(\frac{63} {2} \, \mathrm{m}\)
\(\frac{350} {9} \, \mathrm{m}\)
\(\frac{72} {7} \, \mathrm{m}\)
\(\frac{36} {35}\, \mathrm{m}\)

9000150503

Level: 
C
A pendulum constituted of a rope of the length \(l\) and a body is displaced from it's equilibrium. The force due to gravity on the body \(F_{g} = 20\, \mathrm{N}\). The body is higher by \(h = 10\, \mathrm{cm}\) in the displaced position (comparing to the equilibrium position). The tension in the rope in the displaced position is \(F_{1} = 12\, \mathrm{N}\). Find the length of the rope \(l\). Hint: Using a parallelogram, the force of gravity on the body can be decomposed into a force \(F_{1}\) in the direction of the rope and \(F_{2}\) in the perpendicular direction.
\(25\, \mathrm{cm}\)
\(25\, \mathrm{m}\)
\(6\, \mathrm{cm}\)
\(16\frac{2} {3}\, \mathrm{cm}\)

9000150504

Level: 
C
The object \(y\) is projected using a lens with foci at \(F\) and \(F'\). The focal length of the lens (the distance from the focus to the lens) \(f = 20\, \mathrm{cm}\). The distance from the object \(y\) to the lens \(a = 60\, \mathrm{cm}\). Find the distance from the lens to the image \(y'\).
\(30\, \mathrm{cm}\)
\(600\, \mathrm{cm}\)
\(\frac{20} {3} \, \mathrm{cm}\)
\(25\, \mathrm{cm}\)

9000150505

Level: 
C
The iron support has the shape of the right triangle \(ABC\) with the side \(AB\) of the length \(30\, \mathrm{cm}\) and the hypotenuse \(AC\) of the length \(50\, \mathrm{cm}\) (see the picture). The maximal allowed force \(F_{1}\) on \(AB\) is \(270\, \mathrm{N}\). Find the maximal force \(G\) allowed at the point \(A\). Hint: The load \(G\) at the point \(A\) can be decomposed to the direction of the hypotenuse and the other side of the triangle as shown in the picture.
\(360\, \mathrm{N}\)
\(450\, \mathrm{N}\)
\(540\, \mathrm{N}\)
\(162\, \mathrm{N}\)