C

1103191306

Level: 
C
Find the volume (in liters) of a bucket. The bucket is in the shape of frustum of a cone (see the picture) with the top and bottom diameter of \( 23\,\mathrm{cm} \) and \( 18\,\mathrm{cm} \) and the slant height of \( 17\,\mathrm{cm} \). Round your answer to \( 2 \) decimal places.
\( 5.58\,\mathrm{l} \)
\( 5.65\,\mathrm{l} \)
\( 22.32\,\mathrm{l} \)
\( 22.56\,\mathrm{l} \)

1103191305

Level: 
C
What is the area of a metal plate needed to produce one bucket? The bucket is in the shape of a frustum of a cone as shown in the picture. The top and bottom diameters are \( 23\,\mathrm{cm} \) and \( 18\,\mathrm{cm} \) and the slant height is \( 17\,\mathrm{cm} \). Round your result to \( 1 \) decimal place.
\( 1349.3\,\mathrm{cm}^2 \)
\( 3207.6\,\mathrm{cm}^2 \)
\( 2189.7\,\mathrm{cm}^2 \)
\( 1623.2\,\mathrm{cm}^2 \)

1103191304

Level: 
C
A builders bucket is in the shape of a frustum of a right circular cone as shown in the picture. Find the volume of the bucket with the top and bottom diameter of \( 10\,\mathrm{cm} \) and \( 15\,\mathrm{cm} \) and with the height of \( 18\,\mathrm{cm} \).
\( 712.5\pi\,\mathrm{cm}^3 \)
\( 350\pi\,\mathrm{cm}^3 \)
\( 2023.5\pi\,\mathrm{cm}^3 \)
\( 2850\pi\,\mathrm{cm}^3 \)

1103191303

Level: 
C
A frustum of a pyramid has square ends and the squares have sides \( 18\,\mathrm{cm} \) and \( 6\,\mathrm{cm} \) long, respectively. Calculate the surface area of the frustum if the perpendicular distance between its ends is \( 8\,\mathrm{cm} \).
\( 840\,\mathrm{cm}^2 \)
\( 360\,\mathrm{cm}^2 \)
\( 480\,\mathrm{cm}^2 \)
\( 804\,\mathrm{cm}^2 \)

1103191302

Level: 
C
A frustum of a pyramid has square ends and the squares have sides \( 8\,\mathrm{cm} \) and \( 6\,\mathrm{cm} \) long, respectively. Calculate the volume of the frustum if the perpendicular distance between its ends is \( 12\,\mathrm{cm} \).
\( 592\,\mathrm{cm}^3 \)
\( 9616\,\mathrm{cm}^3 \)
\( 1776\,\mathrm{cm}^3 \)
\( 248\,\mathrm{cm}^3 \)

1003191301

Level: 
C
A frustum of a pyramid has rectangular ends and the sides of the base are \( 8\,\mathrm{cm} \) and \( 6\,\mathrm{cm} \) long. Find the volume of the frustum knowing that the area of the top end is \( 12\,\mathrm{cm}^2 \) and the height of the frustum is \( 5\,\mathrm{cm} \).
\( 140\,\mathrm{cm}^3 \)
\( 100\,\mathrm{cm}^3 \)
\( 420\,\mathrm{cm}^3 \)
\( 1060\,\mathrm{cm}^3 \)

1103212905

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with its bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Find the parametric equations of an intersection line \( p \) of planes \( \alpha \) and \( \beta \), where \( \alpha \) passes through the points \( B \), \( C \) and \( V \), and \( \beta \) passes through the points \( A \), \( D \) and \( V \). What is the measure of an angle \( \varphi \) between the planes \( \alpha \) and \( \beta \). Round \( \varphi \) to the nearest minute.
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=0;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3+t, &\\ z&=6+2t;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)

1103212904

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with a bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the edge \( AD \). Find the standard equation of the plane \( \alpha \) passing through the points \( B \), \( V \) and \( C \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)

1103212903

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Find an angle \( \varphi \) between the plane \( \alpha \) passing through the points \( E \), \( D \) and \( C \) and the straight line \( AF \). Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.
\( \varphi = 30^{\circ} \)
\( \varphi = 15^{\circ} \)
\( \varphi = 45^{\circ} \)
\( \varphi = 60^{\circ} \)

1103212902

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the face \( ABFE \), and let \( K \) and \( L \) be the midpoints of edges \( DH \) and \( CG \) consecutively. Find the standard equation of a plane \( \alpha \) passing through the points \( A \), \( B \) and \( L \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)
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