B

9000045707

Level: 
B
Given a regular pentagon with the side \(a\), find the radius \(\rho \) of the circle inscribed to this pentagon.
\(\rho = \frac{a} {2} \cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
\(\rho = \frac{2a} {\mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = \frac{a} {2\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)

9000039302

Level: 
B
Find \(N\), the number of the turns, as a function of the other variables in the formula for the magnetic induction of a solenoid. \[ B =\mu \frac{NI} {l} \]
\(N = \frac{Bl} {\mu I} \)
\(N = \frac{Bl\mu } {I} \)
\(N = B -\mu \frac{I} {l} \)
\(N = \frac{Bl} {\mu } - I\)

9000045708

Level: 
B
Given a regular hexagon with the side \(a\), find the radius \(\rho \) of the circle inscribed to this hexagon.
\(\rho = \frac{a} {2\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }\)
\(\rho = \frac{2a} {\mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 60^{\circ }\)

9000046405

Level: 
B
A circle is circumscribed to the regular octagon. The perimeter of the octagon is \(16\, \mathrm{cm}\). Find the radius of the circle and round the result to two decimal places. (The regular octagon is a polygon which has eight sides of equal length. The perimeter of the octagon is the sum of the length of all eight sides.) Circle circumscribed to the regular octagon.
\(2.61\, \mathrm{cm}\)
\(1.08\, \mathrm{cm}\)
\(1.41\, \mathrm{cm}\)

9000039304

Level: 
B
Find the focus length \(f\) as a function of the other variables from the following equation relating this distance with object and image distances \(a\) and \(a'\). \[ \frac{1} {f} = \frac{1} {a} + \frac{1} {a'} \]
\(f = \frac{aa'} {a+a'}\)
\(f = \frac{a-a'} {a+a'}\)
\(f = a + a'\)
\(f = \frac{a} {a'}\)

9000039305

Level: 
B
Find \(m_{1}\) as a function of the other variables from the following mixing equation. \[ w_{1}m_{1} + w_{2}m_{2} = w_{3}m_{3} \]
\(m_{1} = \frac{w_{3}m_{3}-w_{2}m_{2}} {w_{1}} \)
\(m_{1} = \frac{w_{3}m_{3}w_{2}m_{2}} {w_{1}} \)
\(m_{1} = \frac{w_{3}m_{3}+w_{2}m_{2}} {w_{1}} \)
\(m_{1} = \frac{w_{2}m_{2}-w_{3}m_{3}} {w_{1}} \)

9000039005

Level: 
B
Find all the values of \(x\) for which the following expression is positive. \[ \frac{2x - 3} {7 - 3x} \]
\(x\in \left (\frac{3} {2}, \frac{7} {3}\right )\)
\(x\in \left (\frac{3} {2},+\infty \right )\)
\(x\in \left (\frac{7} {3},+\infty \right )\)
\(x\in (0,+\infty )\)

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.
\(71.57^{\circ }\)
\(45^{\circ }\)
\(63.43^{\circ }\)