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1103191306

Level:

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 cm

and

18 cm

and the slant height of

17 cm

. Round your answer to

2

decimal places.

5.58 l

5.65 l

22.32 l

22.56 l

1103191305

Level:

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 cm

and

18 cm

and the slant height is

17 cm

. Round your result to

1

decimal place.

1349.3 {cm}^{2}

3207.6 {cm}^{2}

2189.7 {cm}^{2}

1623.2 {cm}^{2}

1103191304

Level:

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 cm

and

15 cm

and with the height of

18 cm

712.5 π {cm}^{3}

350 π {cm}^{3}

2023.5 π {cm}^{3}

2850 π {cm}^{3}

1103191303

Level:

A frustum of a pyramid has square ends and the squares have sides

18 cm

and

6 cm

long, respectively. Calculate the surface area of the frustum if the perpendicular distance between its ends is

8 cm

840 {cm}^{2}

360 {cm}^{2}

480 {cm}^{2}

804 {cm}^{2}

1103191302

Level:

A frustum of a pyramid has square ends and the squares have sides

8 cm

and

6 cm

long, respectively. Calculate the volume of the frustum if the perpendicular distance between its ends is

12 cm

592 {cm}^{3}

9616 {cm}^{3}

1776 {cm}^{3}

248 {cm}^{3}

1003191301

Level:

A frustum of a pyramid has rectangular ends and the sides of the base are

8 cm

and

6 cm

long. Find the volume of the frustum knowing that the area of the top end is

12 {cm}^{2}

and the height of the frustum is

5 cm

140 {cm}^{3}

100 {cm}^{3}

420 {cm}^{3}

1060 {cm}^{3}

1103212905

Level:

A rectangle-based right pyramid

A B C D V

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

α

and

β

, where

α

passes through the points

B

C

and

V

, and

β

passes through the points

A

D

and

V

. What is the measure of an angle

φ

between the planes

α

and

β

. Round

φ

to the nearest minute.

\begin{aligned} p : x & = 3 + t, & φ ≐ 53^{\circ} 8^{'} \\ y & = 3, \\ z & = 6; t \in R \end{aligned}

\begin{aligned} p : x & = 3 + t, & φ ≐ 63^{\circ} 8^{'} \\ y & = 3, \\ z & = 0; t \in R \end{aligned}

\begin{aligned} p : x & = 3 + t, & φ ≐ 53^{\circ} 8^{'} \\ y & = 3 + t, \\ z & = 6 + 2 t; t \in R \end{aligned}

\begin{aligned} p : x & = 3 + t, & φ ≐ 63^{\circ} 8^{'} \\ y & = 3, \\ z & = 6; t \in R \end{aligned}

1103212904

Level:

A rectangle-based right pyramid

A B C D V

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

A D

. Find the standard equation of the plane

α

passing through the points

B

V

and

C

, and calculate the distance of the point

S

from

α

α : 2 y + z - 12 = 0; d = | S α | = \frac{12 \sqrt{5}}{5}

α : 2 x + z - 12 = 0; d = | S α | = \frac{12 \sqrt{5}}{5}

α : 2 y + z - 12 = 0; d = | S α | = \frac{6 \sqrt{5}}{5}

α : 2 x + z - 12 = 0; d = | S α | = \frac{6 \sqrt{5}}{5}

1103212903

Level:

A cube

A B C D E F G H

with an edge length of

2

units is placed in a coordinate system (see the picture). Find an angle

φ

between the plane

α

passing through the points

E

D

and

C

and the straight line

A F

. Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.

φ = 30^{\circ}

φ = 15^{\circ}

φ = 45^{\circ}

φ = 60^{\circ}

1103212902

Level:

A cube

A B C D E F G H

with an edge length of

2

units is placed in a coordinate system (see the picture). Let

S

be the midpoint of the face

A B F E

, and let

K

and

L

be the midpoints of edges

D H

and

C G

consecutively. Find the standard equation of a plane

α

passing through the points

A

B

and

L

, and calculate the distance of the point

S

from

α

α : x + 2 z - 2 = 0; | S α | = \frac{2 \sqrt{5}}{5}

α : x + 2 z - 2 = 0; | S α | = \frac{2 \sqrt{3}}{3}

α : x + 2 y - 2 = 0; | S α | = \frac{2 \sqrt{5}}{5}

α : x + 2 y - 2 = 0; | S α | = \frac{2 \sqrt{3}}{3}

C

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