Triangles

1103021906

Level: 
C
The distance between the places \( A \) and \( C \) on a straight road is \( 300\,\mathrm{m} \). There is a balloon \( B \) above the road between the places \( A \) and \( C \). See the picture. The angles of elevation from the places \( A \) and \( C \) to the balloon \( B \) are \( 20^{\circ} \) and \( 40^{\circ} \) respectively. What is the height \( h \) of the balloon?
\( 76\,\mathrm{m} \)
\( 168\,\mathrm{m} \)
\( 488\,\mathrm{m} \)
\( 523\,\mathrm{m} \)

1003021905

Level: 
B
Calculate the height between two floors if you know that the number of stairs between the floors is \( 16 \), the slope of the staircase is \( 30^{\circ} \) and the depth of a stair is \( 25\,\mathrm{cm} \).
\( \frac{400}{\sqrt3}\,\mathrm{cm} \)
\( \frac{25}{\sqrt3}\,\mathrm{cm} \)
\( 200\,\mathrm{cm} \)
\( 400\,\mathrm{cm} \)

1103021904

Level: 
C
From the highest window of Orava Castle, the angles of depression to the banks of the Orava river are \( 60^{\circ} \) and \( 20^{\circ} \). The height of the window above the river is \( 50\,\mathrm{m} \). What is the width of the river?
\( 108.5\,\mathrm{m} \)
\( 137.4\,\mathrm{m} \)
\( 100.5\,\mathrm{m} \)
\( 125.4\,\mathrm{m} \)

1103021903

Level: 
C
An observer was watching an approaching plane flying at a height of \( 3000\,\mathrm{m} \) in a straight line with constant velocity. At the first moment of measurement the observer saw the plane to be at an angle of elevation of \( 25^{\circ} \). After \( 10 \) seconds the angle of elevation changed to \( 35^{\circ} \). What was the speed of the plane? Round the result to ones.
\( 215\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 2149\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 6576\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 658\,\mathrm{m}\cdot\mathrm{s}^{-1} \)

1003021902

Level: 
B
What is the width of a computer screen if the ratio of its width and height is \( 16:9 \) and the computer has \( 23 \)-inch monitor? Round the result to two decimal places. (\( 1 \) inch=\( 2.54\,\mathrm{cm} \))
\( 50.92\,\mathrm{cm} \)
\( 20.05\,\mathrm{cm} \)
\( 11.28\,\mathrm{cm} \)
\( 28.65\,\mathrm{cm} \)

1103256903

Level: 
C
In isosceles triangle \( ABC \), \( |AB| = 8\,\mathrm{cm} \), \( |BC|=|AC| = 6\,\mathrm{cm} \). Determine what percentage of the triangle area is a circle that is inscribed in it. Round the result to full percentages.
\( 56\,\% \)
\( 48\,\% \)
\( 62\,\% \)
\( 64\,\% \)

1103256902

Level: 
B
The cucumber field has the shape of an isosceles right triangle. The length of its legs is \( 12\,\mathrm{m} \). Rotary sprinklers placed in its vertices have a reach of \( 6\,\mathrm{m} \). Find the area of the field that is not sprinkled with water. Round the result to two decimal places.
\( 15.45\,\mathrm{m}^2 \)
\( 41.10\,\mathrm{m}^2 \)
\( 16.29\,\mathrm{m}^2 \)
\( 15.25\,\mathrm{m}^2 \)

1103021609

Level: 
B
Points \( A \), \( B \) and \( C \) lie on the circle \( k \). The line segment \( AC \) is the diameter of the circle and the lines \( AC \) and \( BC \) contain the angle of \( 60^{\circ} \). Calculate the length of \( AC \) if the length of \( BC \) is \( 10\,\mathrm{cm} \). (See the picture.)
\( 20\,\mathrm{cm} \)
\( 5\sqrt3\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)
\( 2\sqrt3\,\mathrm{cm} \)

1003021607

Level: 
A
Consider a right-angled triangle \( ABC \) with the right angle at the vertex \( C \). Calculate the measure of the angle \( CAB \), if the side \( b=9\,\mathrm{cm} \) and the radius of the circumscribed circle \( r=6\,\mathrm{cm} \). Round the result to one decimal place.
\( 41.4^{\circ} \)
\( 48.6^{\circ} \)
\( 36.9^{\circ} \)
\( 48.2^{\circ} \)