C

1103235604

Level: 
C
The area of the base of a regular hexagonal pyramid is \( 54\sqrt3\,\mathrm{cm}^2 \) and the lateral edge is two times the length of the base edge (see the picture). Find the volume of the pyramid.
\( 324\,\mathrm{cm}^3 \)
\( 108\sqrt3\,\mathrm{cm}^3 \)
\( 972\,\mathrm{cm}^3 \)
\( 216\,\mathrm{cm}^3 \)

1103235603

Level: 
C
The base of a right pyramid is a regular hexagon of side \( 4\,\mathrm{m} \) and its slant surfaces are inclined to the horizontal at an angle of \( 30^{\circ} \) (see the picture). Find the volume.
\( 16\sqrt3\,\mathrm{m}^3 \)
\( 72\sqrt3\,\mathrm{m}^3 \)
\( 48\sqrt3\,\mathrm{m}^3 \)
\( 24\sqrt3\,\mathrm{m}^3 \)

1103235602

Level: 
C
Find the surface area of a right pyramid whose base is a regular hexagon each side of which is \( 6\,\mathrm{cm} \) and height \( 9\,\mathrm{cm} \) (see the picture).
\( 162\sqrt3\,\mathrm{cm}^2 \)
\( 15\sqrt3\,\mathrm{cm}^2 \)
\( 9\left(\sqrt3+6\sqrt{13}\right)\,\mathrm{cm}^2 \)
\( 117\sqrt3\,\mathrm{cm}^2 \)

1103235601

Level: 
C
Find the volume of a right pyramid whose base is a regular hexagon each side of which is \( 6\,\mathrm{cm} \) and height \( 8\,\mathrm{cm} \) (see the picture).
\( 144\sqrt3\,\mathrm{cm}^3 \)
\( 72\sqrt3\,\mathrm{cm}^3 \)
\( 48\sqrt3\,\mathrm{cm}^3 \)
\( 24\sqrt3\,\mathrm{cm}^3 \)

1103021907

Level: 
C
A plane is flying at a speed of \( 900\,\mathrm{km}\cdot\mathrm{h}^{-1} \) and according to the compass the axis of the airplane is still heading westbound. What angle will the flight path of the airplane make with the east-west direction if the south wind starts to blow at a speed of \( 10\,\mathrm{m}\cdot\mathrm{s}^{-1} \)? Round the result to two decimal places.
\( 2.29^{\circ} \)
\( 0.64^{\circ} \)
\( 0.01^{\circ} \)
\( 87.71^{\circ} \)

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: 
C
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: 
C
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=\( 2.54\,\mathrm{cm} \))
\( 50.92\,\mathrm{cm} \)
\( 20.05\,\mathrm{cm} \)
\( 11.28\,\mathrm{cm} \)
\( 28.65\,\mathrm{cm} \)
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