Solved problems for the chapter:

Volume of Solids

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Pyramid for KS4

0.1. Types of pyramids:

0.2. Regular pyramid and irregular pyramids:

0.3. Surface area of a pyramid:

0.4. Volume of a pyramid:

1. Forum

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Prism for KS4

1. Types of prisms

1.1. Vertical and oblique prisms

1.2. Regular prisms

1.3. Equilateral prism

2. Prism surface area

3. Prism volume

4. Forum

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Quadratic inequality

1. Leading coefficient a > 0 (function facing up)

1.1. D > 0 (the function has two real zeros)

1.2. D = 0 (the function has a repeated real zero)

1.3. D < 0 (function has no real zeros)

2. Leading coefficient a < 0 (downward facing function)

2.1. D > 0 (the function has two real zeros)

2.2. D = 0 (function has a repeated real zero)

2.3. D < 0 (function has no real zeros)

3. Forum

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Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:volume_of_solids/3668/

Example 1.

We have bricks with a length of 2.4 dm, a width of 12 cm and a height of 60 mm.

Calculate the area and volume of the brick.
How many bricks can you put in a box 1.2 m long, 12 dm wide and 120 cm high?

Example 2.

A cuboid and a cube have the same volume of 216 cubic centimetres. Calculate the total surface area of the cuboid and cube if the length of the cuboid is and the width is .

Calculate the height of a cuboid, long and wide, if its area is .

Calculate the volume and surface area of the 30 cm high regular four-sided pyramid with a base of 37 cm square.

Calculate the area and the volume of the cuboid in the sketch above and the length of the marked diagonals.

Calculate the width of a cuboid, long and high, if its total surface area is

Calculate the mass of air in a classroom 10 m long, 6 m wide and 4 m high. Note that 20% of the volume of the entire classroom is occupied by furniture. For air density take 1.2 kilograms per cubic meter.

Calculate the length of a cuboid, wide and high, if its total surface area is .

How many litres of water do we need to fill a 25 m long, 18 m wide and 160 cm deep pool? What is the mass of the water in the pool then? Calculate the pressure at the bottom of the pool.

10.

The sketch of a glass aquarium net without a lid is shown above.

Write down the length, width and height of the aquarium.
How many square meters of glass do we need to make such an aquarium?
Can we pour 10 liters of water into an empty aquarium?

11.

Calculate the area of a sphere with volume

12.

What is the volume of a sphere drawn on a cube with edge ?

13.

Calculate the surface area and volume of a square based pyramid with equal edges if the diagonal of the square measures 8 cm.

14.

The parachute has the shape of a semicircle with a radius of . Approximately how much air does it capture?

15.

Three spherical balls have radii , and . If we melt all three into one ball. Calculate its area and volume.

16.

Copper bead has radius . Calculate its mass if ?

17.

Calculate the radius of a sphere equal in volume to a cube with edge .

18.

What is the area of an upright cone that has the same volume as a sphere with area , and the height of the cone is equal to the diameter of the sphere?

19.

The radius of a soccer ball is 12 cm. At night, in the cold, the volume of the ball decreased by 6%. Calculate the new volume and radius of the ball.

20.

Hemisphere of a cylinder have a surface area . What is the area and volume of the composite body if its height is ?

21.

Mom poured 3 spoon of soup into Jerry's plate. The spoon has the shape of a hemisphere with a radius of 3.5 cm. How many decilitres of soup did mom pour for Jerry?

22.

A cuboid with edges , and is used to form a sphere. What is the area and volume of the delineated sphere?

23.

A gold bar in the shape of a square with edges 10 cm, 5 cm and 4 cm is used to make pendants in the shape of a sphere with a diameter of 0.4 cm (the balls are full). At most, how many pendants can they make from one such rod?

24.

In a cube with edge a substitute 27 identical spheres touching each other and the walls of the cube. Calculate the ratio between the volume of the cube and the sum of the volumes of all spheres.

25.

Calculate the area of a sphere with volume .

26.

The body consists of a truncated cone, which has the radii of the base surfaces 10 cm and 5 cm, the height 13 cm, and has a glued hemisphere on the smaller base surface. Calculate the volume and area of the body.

27.

Two spheres are melted into the same metal with surfaces area and into one sphere. What is the radius of the new sphere?

28.

What is the area and volume of a sphere whose radius is r = 6 dm and height h = 8 dm?

29.

The sum of the volumes of two spheres is equal to the volume of a cube with edge . What are the volumes of the spheres if their radii are in the ratio ?

30.

In a cube with edge a = 32cm, draw 8 identical spheres in two layers of 4 so that they touch. What is the ratio of the sum of the volumes of the spheres and the volume of the cube? How does the ratio change if we draw 27 identical spheres in three layers of 9 or 64 identical spheres in four layers of 16 in the same cube?

31.

What is the volume and area of an equilateral cone inscribed with a sphere of radius

32.

Place a styrofoam ball with radius on the water surface. The ball sinks for . Calculate the circumference of a small spherical circle.

33.

A sphere and an equilateral cylinder have the same volume. What is the area of the sphere if the radius of the base plane of the cylinder is ?

34.

Three identical spheres with radius are placed in a cylindrical container having the smallest possible volume and the smallest possible diameter. How much air does the container take up?

35.

In a rectangular box that is 96 cm long, 64 cm wide and 75.2 cm high, we want to stack as many cans as possible. How many cylindrical cans with a base area and a height h = 15 cm can be stacked in this box?

36.

What is the area and volume of an upright cone of a sphere with radius if the height of the cone is equal to the diameter of the base plane?

37.

Calculate the volume and area of the pyramid whose net is given in the figure below, if possible.

38.

Jane made a rice bowl in the shape of a right-angled circular cylinder. He filled it with rice (the bowl was full). Wanting to give it to his mother for her birthday, he covered it with decorative paper. Calculate the area of the cylinder (paper used), if the radius and height of the container is in the ratio 3:6?

39.

We have a given cone with slant height and height . Calculate its total surface area and volume.

40.

We have a given cone having the area of the base and the slant height . Calculate its total surface area and volume.

41.

Tony bought a cone in the store, the slant height of of the cone is , and the opening at the top of the cone is in diameter. Calculate how much ice cream is poured in the cone if the distance from the top of the cone to the top of the ice cream is .

42.

What is the radius r of the base of a cone which is high and has a volume ?

43.

Calculate the angle at the top of the axial section of the cone whose curved surface is obtained by turning three-quarters of a circle with radius .

44.

How much is the radius r of the base area of a cone with total surface area and slant height ?

45.

We have two vertical cones that have the same circular base areas. What is the ratio of the volumes of the cones if the curved surface area of the first cone is times its circular base and the curved surface area of the second is times its circular base?

46.

How much is the slant height l of a cone that is tall and has a circular base area of ?

47.

A vertical cone has the area of its circular base as , and its curved surface area is equal to three times the area of the circular base. What is the volume of such a cone?

48.

What is the slant height of a cone with circular base area of and a curved surface area of .

49.

Calculate the central angle in the curved surface of the unfolded net of a vertical cone with volume and radius of the circular base .

50.

What are the total surface area and volume of a cone having a slant height and a radius of the circular base ?

51.

An high cone has the circumference c of the base area as . Calculate the total surface area and volume of this cone.

52.

A vertical cone, whose circular base is of circumference , has a right triangle for its axial section. Find its total surface area and volume.

53.

Calculate the total surface area and volume of an equilateral cone having an equilateral triangle with area .

54.

What is the central angle of an unfolded equilateral cone?

55.

We have a cone with an axial cross-sectional area and a circular base diameter . Calculate its total surface area and volume.

56.

A equilateral cone has the area of the mantle and the radius of the circular base . Calculate its volume.

57.

What is the volume of a cone with the radius of the circular base and curved surface area that is four times the base area?

58.

Calculate the volume and area of an upright cone high whose slant height and circular base are enclosed as .

59.

Calculate the volume of a a high cone having a circular base area and a curved surface area in the ratio .

60.

Calculate the area of an equilateral cone with volume .

61.

Calculate the slant height, height and volume of an upright cone with total surface area and radius of circular base .

62.

An Iron crate with length , width and height is filled with sand which is then shaken into high upright cone. Calculate the diameter of the circular base of the resulting cone.

63.

At what height should we cut a high metal cone whose diameter of the base surface measures so that both parts will have the same volume?

64.

In the sand pit, they had sand in cone-shaped containers. Before the truck loaded the sand, the sand reached high, and the circumference at the foot measured . How much sand was loaded on the truck if the height of the sand decreased by and the circumference at the base decreased by ?

65.

Calculate the volume and area of the body in the figure.

66.

One of the sides of a squared base pyramid measures , and its height . We also have a cone with the radius of the circular base as and the height . Calculate:

The total surface area of the pyramid and the cone.
what angle is enclosed by the slant height and the base surface of the pyramid.
by what percentage it is necessary to increase the height of the pyramid so that the volumes of both bodies are equal.

67.

The volume of the truncated cone measures and the radius and . Calculate its total surface area.

68.

How much soil can be sprinkled in a container of the shape of an upright truncated cone with radii and and the total surface area of ?

69.

How much sand can be sprinkled in a container of the shape of an upright truncated cone with radii and and side ?

70.

How many litres of water can be poured into a container of the shape of an upright truncated cone with radii and and a coat ?

71.

Kate made a metal base in the shape of a cylinder and glued a wooden rocket to it (see picture). Calculate the total surface area of the resulting body and calculate how much wood is used to make the rocket.

Sketch

72.

Calculate the central angle in the plane of the vertical cone whose curved surface area is four times the base area.

73.

Calculate the center angle into the plane of the unfolded mantle of the cone. The angle at the top of the cone measures and its side measures .

74.

Determine the ratio of the volumes of the cone to the largest regular four-sided pyramid cut from the cone.

75.

Calculate the volume of the cone whose mantle measures and the circumference of the circular base .

76.

Tony made a rattle that has the shape of two joined truncated cones (figure below). Calculate how much rice he put in the rattle if he filled it to .

77.

Calculate the area and volume of a right circular cylinder with the data:

78.

Calculate the surface area and volume of the figure below:

Figure

79.

Calculate the surface area and volume of a cube whose length is .

80.

Given triangle ABC:

Calculate the length |AC| and angle BAC to be denoted as .
Calculate the area of the triangle.
Calculate the surface area and volume of a prism whose base is the given triangle and whose height is 1 m.

81.

Calculate the area of the shaded triangle drawn in a cube with side 7 cm as shown in the figure below:

Figure

82.

Calculate the area and volume of an upright cylinder with the data:

83.

Calculate the length of the diagonal of the side ABEF, the volume and surface area of the figure below.

Figure

84.

Calculate the volume of a cube whose surface area measures .

85.

Calculate the length of the hypotenuse in the right triangle ABC, the volume and surface area of the figure below:

Figure

86.

Find the range of values of m for which the quadratic function is negative for all real x.

87.

If unit cubes are assembled to form the figure below, calculate its surface area and volume.

Figure

88.

The height of an equilateral three-sided pyramid has the same length as the height of the equilateral triangle at the base surface of this pyramid. Calculate the volume of the pyramid if its surface area is .

89.

Calculate the volume and area of the pyramid whose net is in the figure below, if possible.

90.

Calculate the area of a right circular cylinder, given the area of its curve surface, and the volume, .

91.

The edge of a regular four-sided pyramid measures 30 dm and its height is 20 dm. Calculate the area of the lateral surface and the volume of the pyramid.

92.

Calculate the weight of a wooden kitchen counter with length 3 m, width 8.4 dm and height 8 mm. If the density of the wood .

93.

Calculate the area and volume of a regular four-sided pyramid if its axial section is an equilateral triangle with area and its legs are the heights of the opposite side faces.

94.

Calculate the weight of a wooden slab which is 3 m long, 70 cm wide and 4.3 cm thick. Given wood density .

95.

How many unit cubes do we need to add to the figure below to get a cube with side length 4 cm?

Figure

96.

Calculate the surface area of a triangular prism with base side lengths a = 12.7 cm, b = 14.5 cm and c = 21 cm when its volume is .

97.

A right-angled triangular prism has two of its base lengths as 4 cm and 7 cm. If the third side is the hypotenuse and the prism is 10 cm high, calculate its surface area and volume.

98.

The pyramid has a square with a side 25 cm as its base, and its side edge measures 20 cm. Calculate the slant height, the angle between the base and side surfaces, and the surface area of the pyramid.

99.

The surface area of a cube and a cuboid is equal. If the side lengths of the cuboid are a = 12.5 cm, b = 4.5 cm and h = 3.5 cm, calculate the length of the cube and its volume.

100.

Calculate the height of a right circular cylinder given it's curve surface area, and its total surface area, .

101.

The side lengths of a cuboid are in the ratio a: b: h = 7: 3: 11. Calculate the surface area and volume of the cuboid if the diagonal of the rectangle with sides b and h is 15 cm long.

102.

Axial section of a wooden pillar in the shape of a cylinder, height 130 cm, has a diagonal length 170 cm. Calculate the area and volume of the column.

103.

The side of the base of a cubic prism measures 23 cm and its surface area is . Calculate the volume of the prism, the diagonal of the base and the surface diagonal.

104.

Calculate the area of a cylindrical box whose volume measures and the area of the curve surface is equal to the area of the base surface.

105.

The axial section area 2.1 m of a high concrete column having the shape of a right circular cylinder measures . What is its area and volume?

106.

Calculate the total surface area and volume of the cylinder formed by a square with side 17 cm rotate about a symmetrical axis by .

107.

Calculate the total surface area and volume of the cylinder formed if a rectangle with sides 13 cm and 10 cm, rotate around the longer side for .

108.

What is the ratio of the surface area of a cuboid to the surface area of a cube if the cuboid dimensions are 300 cm, 2.5 dm and 0.4 m and the volume of the cube and the cuboid are equal?

109.

Calculate the volume and surface area of a four-sided pyramid that has a square with side 8 cm as its base. The side surface is enclosed with the base surface at an angle of .

110.

Calculate the area and volume of a four-sided pyramid with a side edge s = 40.2 cm. The base is a rectangle with sides a = 43 cm and b = 20 cm.

111.

What is the length of the side edge of a four-sided pyramid with surface area of if the base surface is a square with side 13 cm?

112.

A 12 cm high triangular prism has its base triangle sides measure a = 17 cm, b = 10 cm and c = 12 cm. Calculate the surface area and volume of the prism.

113.

A triangle with sides 15 cm, 22 cm, and 24 cm is the base of a pyramid of height 13 cm. Calculate the side edge s.

114.

Calculate the surface area of a three-sided pyramid with the lateral area of . The side edges are perpendicular to each other, and the base is an equilateral triangle.

115.

Calculate the base area and area of the diagonal section of a cube with volume .

116.

A regular three-sided pyramid with a height of length 11 cm has a volume of . Calculate the length of the base edge of the pyramid.

117.

On the table stands a right circular cylindrical vessel with an inner diameter 30 mm. Calculate by how much the water level rises if we immerse a cube with an edge in this container 14 mm?

118.

Calculate the length of the base edge of a regular four-sided pyramid with height 11 cm and volume .

119.

Calculate the surface area of a rectangular-based pyramid with a volume of if the base measures 10 cm by 4 cm.

120.

Calculate the base edge of a regular six-sided based pyramid with height 11 cm and a volume of .

121.

Fill in a right circular cylinder with the largest possible volume from an equilateral three-sided prism. Calculate the ratio of the volumes and areas of both bodies.

122.

A rectangle with sides 11 cm and 4.5 cm is the base of a pyramid with a side edge 23 cm. Find the surface area and volume of the pyramid.

123.

Calculate the length of the body diagonal of the cube obtained by melting a metal regular four-sided pyramid with a base edge 4 cm and lateral area of .

124.

The coordinates of a rectangle are given by the points A (1,1), B (7,1), C (7,3) and D (1,3 ).

Draw the figure in a coordinate system and calculate the perimeter of the rectangle ABCD.
Point T lies on side AB, so that ratio , and point S bisects side BC. Draw the points T and S in the given coordinate system and calculate the length of the line TS.
Rectangle ABCD represents the body surface area of the prism with an equilateral triangle base. If the height of the prism is h = 2, calculate the volume of the prism.

125.

A four-sided pyramid has a square with lenght 15 cm as its base surface. Calculate the surface area and volume of the pyramid if the angle between the base and side edge is .

126.

The regular four-sided pyramid has a base edge of length 9 dm, and its side edge measures 13.6 dm. Calculate the area of the lateral area and the volume of the pyramid.

127.

A three-sided pyramid with volume has a triangle with sides 22 cm, 48 cm and 66 cm as its base. Calculate its height.

128.

A cuboid has the same surface area as a cube with a body diagonal . The lengths of side of the cuboid are in the ratio a: b: c = 2: 1: 3. Calculate the volume of the cuboid.

129.

A right-sided pyramid has an equilateral triangle with side 15 cm as the base surface. How much does the side edge of this pyramid measure if the lateral area is 10 times the area of the base? Also, calculate the angle between the base and side edges.

130.

Calculate the surface area and volume of the figure below:

Figure

131.

A trapezium with sides a = 21 cm, b = 19 cm, c = 10 cm and d = 12 cm, represents the base of a four-sided prism. If the rectangular surface area of the prism is twice its base area, calculate the surface area and volume of the prism.

132.

A triangular prism is formed in a cube prism of side a = 11 cm and height h = 12 cm. What percentage does the volume of the three-sided prism represent of the four-sided prism?

133.

The base side of an oblique four-sided prism is a square which measures 5 cm. The angle between the base and side surfaces . Calculate the surface area and volume of this prism if its slant side measures 15 cm.

134.

The pyramid net in the figure below is to be cut out. Calculate the length of the edge of the paper if the volume of the pyramid is

135.

Calculate the angle between the base and side edges of a regular three-sided pyramid of volume if its side edges are perpendicular to each other.

136.

The lateral area of a regular four-sided pyramid measures . Calculate the volume of this pyramid if the area of the base area is equal to one third of the lateral area.

137.

A candy box shape is a regular six-sided prism. The base side length of the prism is 6 cm and 5 cm high as shown in the figure below:

Figure 1

Calculate the area of the base of the prism and the size of the marked angle in the figure.
Calculate the surface area of the prism.
If the total volume of candies in the box is approximately , calculate the proportion of the volume occupied by the candies in the box.

138.

What is the volume of a four-sided pyramid with a lateral area of , if its base is a square with side 6 cm?

139.

The figure below shows three-sided prism ABCDEF. , angle ABC and . If the height of the prism is 56 cm,

Figure 1

Calculate the length of the lines BD.
Calculate the volume of the prism.
Calculate the triangular surface area of the prism.

140.

What is the height of a regular three-sided pyramid with the same volume as a cube with the surface area of , If the base edge of the pyramid is equal to three-quarters of the side diagonal of the cube.

141.

The figure is made of two identical cubes with side 15 cm, as shown below. If the distance between the points and is 14 cm.

Calculate the volume of the figure.
Calculate surface area of the figure.
Calculate the distance between the points and .

Figure 1

142.

What is the ratio of the volume of a regular four-sided pyramid to the volume of a regular four-sided prism having the same base edge and surface area, if the base edge of the prism is three times its height?

143.

The base of a prism is a rhombus with diagonals and as shown in the figure below. If the diagonal of the side surface measures , calculate the surface area of the prism.

Figure

144.

Matthias is making a model of a regular six-sided prism, which he will make from a 20 cm high cylinder with a volume of . Calculate the length of the base edge of the largest possible formed regular hexagonal prism .

145.

A right four-sided pyramid has a base edge of length 2.5 dm, and its side edge is 3 times longer. Calculate:

the angle between the base and the side surfaces,
the angle between the base and side edges,
the angle between the side edge and the base surface,
volume of the pyramid.

146.

What is the surface area of a regular six-sided pyramid with a base edge a = 12 cm and height 30 cm?

147.

What is the surface area of a regular six-sided pyramid with a base edge a = 10 cm and a side edge s = 15 cm?

148.

What is the surface area of a regular six-sided pyramid with height h = 5 cm and side edge s = 15 cm?

149.

Two similar right circular cylinder are of heights 16 cm each. If the the first one with inner diameter of 4 cm is filled with a liquid and the liquid is then poured into the second cylinder of inner diameter 6 cm, find the depth of the liquid in the second container.

150.

A rod with a diameter of and a height of is trimmed so that its diameter is reduced by . By how much cm should we reduce the height h of the original rod to have the same volume as the reduced diameter?

151.

The base of a pyramid is a regular hexagon and is three times smaller than the lateral surface area of the pyramid. Calculate the total surface area and volume of the pyramid if the length of the base edge is 10 cm.

152.

Calculate the volume of a regular six-sided pyramid forming an isosceles triangle with a cross sectional base of 10 cm and arm 16 cm.

153.

John got a drawn and outlined cylinder 5 cm high, three-sided prisms in his math class for homework. The edges of the base plane, represented by a right-angled triangle, measured a = 6 cm, b = 2.6 cm and c = 4.2 cm. He also had to determine the ratio of the areas and volumes of the drawn cylinders.

154.

How high is a regular hexagonal pyramid, whose base edge measures 5 cm and the lateral surface area is .

155.

A wooden regular eight-sided pyramid, 30 cm high with a base edge 10 cm, was transformed into the largest possible regular four-sided pyramid. Calculate what percentage of the volume of the octagonal pyramid the volume of the newly formed pyramid is.

156.

The radius of the circumscribed circle to the base plane of a regular octagonal pyramid measures 12 cm. Calculate the total surface area and volume of the pyramid if it extends 48 cm high.

157.

What is the total surface area and the volume of a regular octagonal pyramid with a base edge of 8 cm and a side edge that is twice as long as the base?

158.

Four-equilateral-sided pyramids with edges of 2.5 cm were added to each surface of a cube of edge a = 5 cm. Calculate the total surface area and volume of the resulting body and what percentage of the total volume is the volume of the added pyramids?

159.

From a wooden regular six-sided prism, a pyramid was carved out with the same base surface as the prism and with a height equal to one third of the base edge of the prism. Calculate the volume and surface area of the resulting wooden body if the base edge of the prism is 15 cm and its height is twice as large.

160.

Calculate the volume and area of a regular hexagonal pyramid with the base edge . The area of the base surface is equal to a quarter of the lateral area.

161.

Calculate the area of the polyhedron in the figure below.

162.

Calculate the area of the polyhedron in the figure below.

163.

Calculate the the volume of a three-sided frustum of a pyramid with a height of 12 cm and the base edges and . The base planes are equilateral triangles.

164.

The total surface area of a right -angled circular cylinder measures . Calculate the volume of the cylinder if the diameter is 3 cm smaller than the height of the cylinder. Also determine the axial cross-sectional area of the cylinder.

165.

We round 30 m copper wires on a coil. What is the diameter of this wire if we used 10 kg copper for it, the density of which is .

166.

The radius of a cylinder is 7 cm and it's volume is . calculate the number of times the area of the base surface is smaller than the right-angled circular cylindrical.

167.

A tree trunk of length 6m and diameter of 6.5dm is cut down from the forest. A large possible beam (four -sided prism) will be made from it. Calculate the percentage of the beam and what percentage is waste.

168.

A round (Cylindrical) birthday cake with a diameter of 24 cm and height 11 cm is cut into different pieces. Calculate the volume and area of a piece of cake with a centred angle

169.

What is the volume of a three-sided frustum of a pyramid with a side edge of 25 cm and the base edges and . The top and base surfaces are equilateral triangles.

170.

A regular triangular frustum of a pyramid that is 30 cm high has a base edge with lengths and . Calculate its surface area and volume.

171.

Mom poured 1.5 l juice into a jug of right-angled circular cylinder shape. How high does the juice reach in a 2.75 l jug that is 35 cm high?

172.

Find the area and volume of the frustum of a pyramid, which has a square with sides , as its base and its height measures 7 cm.

173.

The cylindrical wine container is filled to a height of 63 cm. How many cups of wine can be poured if the radius of the cup is 5 cm and its height 16 cm? Pour the cup up to 1 cm under the edge. The barrel of wine is 1.3 m high and has a diameter of 90 cm.

174.

Calculate the surface area and volume of frustum of a pyramid having a square with sides and and a lateral area of .

175.

A flower pot in the shape of a frustum of a four-sided pyramid with a rectangular base area was given. The smaller base area has sides 5 dm and 3 dm long, while the larger base has both sides that are 30% longer than the lower smaller area. Both surfaces are connected by an edge of length 6 dm. Calculate how many liters of soil go into the trough.

176.

Jessica makes a wooden model of a right-angled circular cylinder at school, the diagonal of the axial section is inclined towards the base surface by , and the area of the axial section is . Calculate how many liters of paint it will use per cylinder if we use approximately on .

177.

How much is the volume of an isosceles oblique cylinder whose axial cross section has a plane . The vertical angle in the axial section is

178.

Calculate the volume and area of the largest regular four-sided prism inscribed in an equilateral cylinder with a diameter of 14 cm.

179.

Calculate the area of the figure below:

180.

The base face of an upright pyramid is a rectangle with sides and , and the height of the pyramid is 8. Sketch and mark as , the angle between the side margin and the base face. Calculate the volume of the pyramid and the magnitude of the angle to a tenth of a degree.

181.

A three-sided pyramid is given.

In triangle with data

and the radius of the triangle of the delineated circle , the point is the center of this triangle of the delineated circle. Calculate the scalar product .
Express the vectors and as a linear combination of vectors

Prove that the equality holds.
Let the pyramid be equal. Prove that the opposite edges and are perpendicular to each other.

182.

The height of a pyramid is 8 cm, the length of the side edge is 15 cm. Draw a sketch of the pyramid and calculate the value of the angle marked between the base surface and the side edge.

183.

The length of the base edge of the regular 4 - side of the pyramid is 4.2 m. The side face of the pyramid is inclined towards the base face by an angle . Draw a sketch of the pyramid, mark the angle and calculate the volume of the pyramid.

184.

James has 5 cylindrical oak wood. The length of an individual wood is 3.5 m and the radius is 0.25 m. How many euros does he get for wood if the price of oak wood is 86 euros per cubic metre?

185.

The base of an upright pyramid ABCDV is a rectangle with sides | AB | = a = 60 cm and | BC | = b = 80 cm. The side edge of the pyramid measures 1.3 m.

Sketch the pyramid and a sketch of its net.
Calculate the volume of the pyramid.
Calculate the area of the side plane BCV.

186.

We have two empty cylindrical tanks standing on their base planes.

The first tank has the shape of an upright cylinder with a radius of . It is filled to two-thirds full with apple juice. Calculate the height of the tank. Round the result to the nearest tenth of a decimetre.
The second tank has the shape of an equilateral cylinder (the axial cross-section is square). Pour of apple juice into it and fill it to the top. Calculate the radius of the tank. Round the result to the nearest tenth of a decimetre..

187.

A regular four-sided pyramid has an edge of the base plane of length 2. The adjacent side edges of the pyramid are enclosed at an angle of .

Express the surface area of the pyramid in terms of .
Calculate the value of , where the volume of the pyramid is .
Show that the volume of the pyramid is .
Let . Show that at this angle the opposite side edges of the pyramid are perpendicular.

188.

In the rectangles appearing in the following tasks, we denote and .

The circumference of the rectangle is 880 cm. Calculate the sides a and b so that the area of the rectangle is the largest.
The rectangle (a = 8 cm, b = 3 cm) is rotated by around an axis outside the rectangle that is parallel to the side and away from it by d = 2 cm. Calculate the volume and area of the spindle. The result should be accurate.
Point M lies on side BC so that | BM | : | MC | = 1: 3. The intersection of the line DM with the diagonal AC is denoted by P. In what proportion does the point P diagonal AC divide? Write down the ratio | AP | : | PC |.

189.

The base of an upright four-sided pyramid is a square with a side of length 2 dm. The height of the pyramid is 6 dm.

Sketch and calculate the volume of the pyramid.
Calculate the area of the pyramid.
Calculate the length of the side edge of the pyramid.

190.

Alexander bought a cylindrical bottle with a radius of 1.5 cm and a height of 15 cm.

He poured 30ml of perfume into the bottle. How high above the bottom is the level of the perfume?
The bottle was in a square shape. Calculate the possible area of the smallest packaging.
After a month, Alexander consumed 15% of the perfume. How high was the perfume level after one month?

191.

Points are given in the plane

How far is the point C away from the line through the points A and B? The result should be accurate.
Calculate the equation of the line of the beam of the center of gravity per side AC of the triangle ABC and the size of the angle at the angle A.
Calculate the vertex D so that the quadrilateral ABCD is a parallelogram.
Calculate the volume of the vortex created by rotating the parallelogram ABCD around the side AB.

192.

In a regular four-sided pyramid, the lateral height is longer than the height h of the pyramid. Calculate the exact length of the base edge of the pyramid if the volume measures .