Triangles

9000038707

Level: 
C
The box is on the slope as in the picture. The length of the slope is \(l = 2\, \mathrm{m}\) and the height is \(h = 1.2\, \mathrm{m}\). The forces acting on the box are the force of gravity \(\vec{F_{G}}\) and the friction \(\vec{F_{t}}\). The force of gravity can be replaced by two components \(\vec{F_{1}}\) and \(\vec{F_{n}}\). (The force \(\vec{F_{1}}\) is parallel to the slope and \(\vec{F_{n}}\) is perpendicular to the slope.) The friction \(F_{t}\) is given by the formula \(F_{t} = fF_{n}\) where \(f\) is the coefficient of the friction. Consider the standard acceleration of gravity \(g = 10\, \mathrm{m\, s^{-2}}\). Find the minimal value for the coefficient of the friction \(f\) to ensure that the box does not move with an acceleration.
\(f = 0.75\)
\(f = 0.6\)
\(f = 0.65\)
\(f = 0.7\)
\(f = 0.55\)
\(f = 0.8\)

9000036102

Level: 
C
Three forces act on the same body in the same point and the total force on the body is zero (the forces cancel). The first two forces are \(8\, \mathrm{N}\) and \(10\, \mathrm{N}\) and the angle between these forces is \(55^{\circ }\). Find the third force.
\(16\, \mathrm{N}\)
\(15\, \mathrm{N}\)
\(17\, \mathrm{N}\)
\(18\, \mathrm{N}\)

9000036103

Level: 
C
Three forces \(F_{1}\), \(F_{2}\) and \(F_{3}\) act on the same body in the same point and the total force on the body is zero (the forces cancel). The first two forces are \(F_{1} = 8\, \mathrm{N}\) and \(F_{2} = 10\, \mathrm{N}\) and the angle between \(F_{1}\) and \(F_{2}\) is \(55^{\circ }\). Find the angle between \(F_{3}\) and \(F_{1}\). Round your answer to the nearest degrees.
\(149^{\circ }\)
\(125^{\circ }\)
\(55^{\circ }\)
\(30^{\circ }\)

9000036106

Level: 
C
Two straight roads go off from the crossing. The angle between directions of the roads is \(52^{\circ }18'\). A significant tree is on the first road in the distance \(250\, \mathrm{m}\) from the crossing. A rock with a beautiful view is on the second road in the distance \(380\, \mathrm{m}\) from the crossing. Find the direct distance (length of a line segment) from the rock to the tree and round your answer to nearest meters.
\(301\, \mathrm{m}\)
\(411\, \mathrm{m}\)
\(568\, \mathrm{m}\)
\(629\, \mathrm{m}\)

9000035001

Level: 
B
The angle of elevation of a straight road is \(3^{\circ }30'\). The distance between two places measured along the road is \(2\, \mathrm{km}\). For these places, find the difference in altitudes, i.e. the vertical distance, and round the result to the nearest meter. (See the picture.)
\(122\, \mathrm{m}\)
\(276\, \mathrm{m}\)
\(98\, \mathrm{m}\)
\(49\, \mathrm{m}\)

9000035007

Level: 
B
A roof gable has the shape of an isosceles triangle with the base of \(14\, \mathrm{m}\). The angle between the roof and the horizontal direction is \(31^{\circ }\). Find the height of the gable. Round your result to one decimal place.
\(4.2\, \mathrm{m}\)
\(5.9\, \mathrm{m}\)
\(3.6\, \mathrm{m}\)
\(11.2\, \mathrm{m}\)

9000035004

Level: 
B
The triangle \(ABC\) has the angle \(\beta = 59^{\circ }\) and the side \(a = 14\, \mathrm{cm}\). Find the altitude \(v_{c}\) (the line segment which is perpendicular to the side \(c\) and joins the vertex \(C\) with the side \(c\)) and round to the nearest centimeters.
\(12\, \mathrm{cm}\)
\(7\, \mathrm{cm}\)
\(10\, \mathrm{cm}\)
\(23\, \mathrm{cm}\)

9000035006

Level: 
B
A ladder of the length \(15\, \mathrm{m}\) leans against a wall. The angle between the ladder and the horizontal direction is \(70^{\circ }\). Find the height of the top of the ladder and round your answer to the nearest meters.
\(14\, \mathrm{m}\)
\(13\, \mathrm{m}\)
\(16\, \mathrm{m}\)
\(15\, \mathrm{m}\)

9000035003

Level: 
B
The tree of the height \(12\, \mathrm{m}\) is observed from the place horizontal with the base of the tree. The angle of elevation is \(10^{\circ }\). Find the distance of the observer from the base and round to the nearest meters.
\(68\, \mathrm{m}\)
\(2\, \mathrm{m}\)
\(12\, \mathrm{m}\)
\(48\, \mathrm{m}\)

9000035008

Level: 
B
Sun shines to the road at the angle \(53^{\circ }22'\). An electric column near the road casts the shadow of the length \(4.5\, \mathrm{m}\). Find the height of the column and round your answer to the nearest meters.
\(6\, \mathrm{m}\)
\(3\, \mathrm{m}\)
\(4\, \mathrm{m}\)
\(5\, \mathrm{m}\)