Triangles

1003021701

Level: 
A
Interior angles of a triangle \( ABC \) are in the ratio \( \alpha:\beta:\gamma=2:4:6 \). Calculate the measures of these angles.
\( \alpha=30^{\circ};\ \beta=60^{\circ};\ \gamma=90^{\circ} \)
\( \alpha=20^{\circ};\ \beta=40^{\circ};\ \gamma=60^{\circ} \)
\( \alpha=15^{\circ};\ \beta=30^{\circ};\ \gamma=135^{\circ} \)
\( \alpha=90^{\circ};\ \beta=60^{\circ};\ \gamma=30^{\circ} \)

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.
\(360\, \mathrm{N}\)
\(450\, \mathrm{N}\)
\(540\, \mathrm{N}\)
\(162\, \mathrm{N}\)

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.
\(\frac{63} {2} \, \mathrm{m}\)
\(\frac{350} {9} \, \mathrm{m}\)
\(\frac{72} {7} \, \mathrm{m}\)
\(\frac{36} {35}\, \mathrm{m}\)

9000124503

Level: 
C
A tall radio mast is attached by several cables. The length of each cable is \(30\, \mathrm{m}\) and all cables are attached \(2\, \mathrm{m}\) under the top of the mast. The second end of the cable is anchored to the ground. The cable is in the height \(6\, \mathrm{m}\) if measured directly above the point which is in the distance \(8\, \mathrm{m}\) from the point where the cable is anchored to the ground. Find the height of the mast.
\(20\, \mathrm{m}\)
\(24\, \mathrm{m}\)
\(22.5\, \mathrm{m}\)
\(24.5\, \mathrm{m}\)

9000124504

Level: 
C
A force due to gravity on a body is \(1\: 800\, \mathrm{N}\). This body has to be lifted to the height \(50\, \mathrm{cm}\) using a slope. The maximal force which can be used to lift the body is \(600\, \mathrm{N}\). Neglect the friction and find the minimal length of the slope required to accomplish this task.Hint: The force due to gravity can be decomposed into two directions. The normal force \(F_{1}\) is compensated by the reaction of the slope. The force \(F_{2}\) parallel to the slope is required to overcome if we wish to lift the body (see the picture).
\(\frac{3} {2}\, \mathrm{m}\)
\(\frac{2} {3}\, \mathrm{m}\)
\(\frac{1} {6}\, \mathrm{m}\)
\(\frac{20} {9} \, \mathrm{m}\)

9000124501

Level: 
C
Similar triangles can be used to estimate the distance from a distant object of a given width. Consider a door of the width \(85\, \mathrm{cm}\). A man stands in an unknown distance from the door and holds a thin pencil vertically in his arm in the distance \(35\, \mathrm{cm}\) from his face. If he closes the left eye, the right eye, the pencil and the left side of the door are aligned in one line. In a similar way, his left eye, the pencil and the right hand side of the door are also aligned in one line, which is apparent when closing the right eye. Assuming the distance \(6\, \mathrm{cm}\) between his eyes, estimate the distance from the man to the door. Give your answer in meters and round to one decimal place.
\(5.3\, \mathrm{m}\)
\(5.0\, \mathrm{m}\)
\(0.5\, \mathrm{m}\)
\(4.5\, \mathrm{m}\)

9000124505

Level: 
C
The picture shows the virtual image \(y'\) of the object \(y\) as created by a concave lens. The points \(F\) and \(F'\) are focal points of the lens. The distance from the lens to each of the focal points is \(20\, \mathrm{cm}\). The object \(y\) is \(25\, \, \mathrm{cm}\) height and it is in the distance \(50\, \mathrm{cm}\) from the lens. Find the height of the virtual image \(y'\).
\(\frac{50} {7} \, \mathrm{cm}\)
\(10\, \mathrm{cm}\)
\(\frac{50} {3} \, \mathrm{cm}\)
\(\frac{175} {2} \, \mathrm{cm}\)