Metric Properties

9000120302

Level: 
A
A cuboid has sides \(a = 5\, \mathrm{cm}\), \(b = 8\, \mathrm{cm}\), and \(c = \sqrt{111}\, \mathrm{cm}\). Find the length of the cuboid’s space diagonal \(u\) (see the picture).
\(10\sqrt{2}\, \mathrm{cm}\)
\(\sqrt{222}\, \mathrm{cm}\)
\(20\, \mathrm{cm}\)
\(2\sqrt{10}\, \mathrm{cm}\)
\(5\sqrt{7}\, \mathrm{cm}\)

9000045709

Level: 
A
Let \(\omega \) be the angle between the solid diagonal of a box and the base of this box. Find the expression which allows to find \(\omega \).
\(\mathop{\mathrm{tg}}\nolimits \omega = \frac{\sqrt{2}} {2} \)
\(\cos \omega = \frac{\sqrt{2}} {2} \)
\(\sin \omega = \frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits \omega = \frac{\sqrt{2}} {2} \)

9000046409

Level: 
B
The base of a pyramid is a square with the side of \(2\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). Find the angle between the lateral side of the pyramid and the base. Round your result to two decimal places.
\(75.96^{\circ }\)
\(70.52^{\circ }\)
\(79.98^{\circ }\)

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