B

9000046408

Level: 
B
Consider a cone of base radius \(r\) and a special shape: the shape is such that the volume of the cone is related to the base radius by the formula \(V =\pi r^{3}\). Find the angle between the side of the cone and the base. Round your answer to two decimal places.
\(71.57^{\circ }\)
\(45^{\circ }\)
\(63.43^{\circ }\)

9000045702

Level: 
B
Given a right triangle \(ABC\) (see the picture). Find the valid relation between the angle \(\alpha \) and the sides of the triangle.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{a} {c}\)
\(\sin \alpha = \frac{a} {c}\)
\(\cos \alpha = \frac{b} {a}\)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \frac{b} {a}\)

9000045710

Level: 
B
Find the length \(l\) of a latitude at \(50^{\circ }\) N. (Use \(R\) for the radius of the Earth.)
\(l = 2\pi R\cos 50^{\circ }\)
\(l = 2\pi R\sin 50^{\circ }\)
\(l = 2\pi R\mathop{\mathrm{tg}}\nolimits 50^{\circ }\)
\(l = 2\pi R\mathop{\mathrm{cotg}}\nolimits 50^{\circ }\)

9000045703

Level: 
B
Given a right triangle \(ABC\) with the right angle at $C$ and an altitude $v$ (see the picture). Find the valid relation between the angle \(\alpha \) and the lengths in the triangle.
\(\sin \alpha = \frac{v} {b}\)
\(\sin \alpha = \frac{v} {c}\)
\(\sin \alpha = \frac{a} {v}\)
\(\sin \alpha = \frac{c} {a}\)

9000045704

Level: 
B
Given a right triangle \(ABC\) with the right angle at $C$ and an altitude $v$ (see the picture). Find the valid relation between the angle \(\beta \) and the lengths in the triangle.
\(\sin \beta = \frac{v} {a}\)
\(\mathop{\mathrm{tg}}\nolimits \beta = \frac{a} {v}\)
\(\cos \beta = \frac{v} {a}\)
\(\mathop{\mathrm{tg}}\nolimits \beta = \frac{v} {a}\)

9000046403

Level: 
B
Consider an isosceles triangle, i.e. the triangle with two sides of equal length. The length of the third side is \(4\, \mathrm{cm}\). One of the interior angles is \(120^{\circ }\). Find the area of this triangle.
\(\frac{4\sqrt{3}} {3} \, \mathrm{cm}^{2}\)
\(4\sqrt{3}\, \mathrm{cm}^{2}\)
\(\frac{8\sqrt{3}} {3} \, \mathrm{cm}^{2}\)

9000046506

Level: 
B
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ \sin 2x =\mathop{\mathrm{tg}}\nolimits x \]
\(2\sin x\cdot \cos x = \frac{\sin x} {\cos x}\)
substitution \( 2x = z\)
\(\sin x = \frac{\mathop{\mathrm{tg}}\nolimits x} {2} \)
\(\cos ^{2}x -\sin ^{2}x =\mathop{\mathrm{tg}}\nolimits x\)

9000046404

Level: 
B
The parallelogram has sides of the length \(5\, \mathrm{cm}\) and \(4\, \mathrm{cm}\) (see the picture). The area of this parallelogram is \(S = 10\sqrt{2}\, \mathrm{cm}^{2}\). Find the measure of the smaller of the interior angles.
\(45^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)