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Test 5, Geometry
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Question 1 of 10
1. Question
What is the volume of the cornice with a length of 1 m? (A cornice, оr “ledge”, is generally any horizontal decorative molding that crowns a building or furniture element.) The crosssection shape of the cornice is shown in the figure below. ED=3 cm, radius FE=4 cm, radius OS=5 cm, side BC=16 cm, AB=3 cm.
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Solution: A crosssectional area of the cornice is 16×3+ π×5^{2}/4+7×9π×4^{2}/4≈118 cm^{2}. The length of the cornice is 100 cm, so its volume is 118×100=1.18×10^{4} cm^{3}.
Answer: ≈12 × 10³ cm³

Question 2 of 10
2. Question
A tank for liquid consists of a hemisphere with radius of 0.25 m and a cylinder with the same base radius. How high should the cylindrical part be if the whole volume of the tank is equal to 0.25 m^{3}?
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Solution: The volume of the cylindrical part is , while the volume of the half sphere is ≈0.033 m^{3}. The total volume is 0.25 m^{3}. Therefore, h = (0.250.033) / (π × 0.25^{2}) ≈1.1 m.
Answer: 1.1 m.

Question 3 of 10
3. Question
A container for liquid consists of a hemisphere of radius R and a cylinder with the same base radius. At what ratio between the radius R and the height of the cylinder h is the volume of liquid in the cylindrical portion equal to the volume of the liquid in the hemisphere?
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Solution: The volume of the cylindrical part is, while the volume of the hemisphere is . According to the problem, they should be the same. Therefore,
Answer: h=2R/3

Question 4 of 10
4. Question
A crystal has the shape of two regular quadrangular pyramids connected by their base planes. A side of the common base is 3.5 cm long, and the distance between the tops of the pyramids are equal to 5 cm. Find the volume of the crystal.
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Solution: The crystal consists of two pyramids. The base plane of both pyramids is the same plane with an area equal to 12.25 cm. The height of one pyramid is 2.5 cm. A volume of one pyramid is (1/3) × 12.25 × 2.5 ≈10.2 cm^{3}. Therefore, the volume of the crystal from two identical pyramids is equal to 20.4 cm^{3}.
Answer: 20.4 cm³

Question 5 of 10
5. Question
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Question 6 of 10
6. Question
A cylindrical water tank has an inner diameter D_{1} equal to 1 m, an outer diameter D_{2} = 1.1 m, and a height H =3 m. How much more paint is needed to paint the outer surface of the tank than the inner surface if 1 m^{2} requires Q=100 g of paint?
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Solution: The difference between the amounts of paint required to paint the outer and the inner surfaces of the tank is equal to π×H×(Ɒ_{2}Ɒ_{1})×Q≈90 g.
Answer: ≈90 g.

Question 7 of 10
7. Question
One of the most spectacular ancient structures, the pyramid of Cheops has the shape of a square pyramid with a height OS of 146.7 m and a lateral edge AS = 219.2 m. Find the volume of the pyramid.
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Solution: To determine the base edge AB, we find the value of AO using the Pythagorean Theorem. Here, , and, therefore, AO≈162.9 m. Triangle AOB is a right and isosceles triangle. Therefore 2AO^{2} = AB^{2}, or AB=AO. Thus, AB ≈ 230.4 m. The height SG of the triangle ASB is ; on the other hand, 2 m. Thus, SG≈186.5 m, and the total lateral surface area of the pyramid is 4 × 230.4 × 186.5 / 2 ≈ 85,939 m^{2}. The volume of the pyramid is (1/3) × 230.4 × 230.4 × 146.7 ≈ 2,595,815 m³
Answer: The pyramid’s lateral area is ≈ 85,939 m^{2}, and its volume is ≈ 2,595,815 m^{3}.

Question 8 of 10
8. Question
If a square pyramid is sliced by a plane EFGM parallel to the base ABCD of the pyramid and a plane bisects the altitude OK of the pyramid, what percentage of the volume of the pyramid (the shaded part in the attached figure) lies below the slicing plane? Keep in mind that ON=NK.
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Question 9 of 10
9. Question
A first cylindrical cup with a diameter of D_{1}= 8 cm and a height of h_{1}= 10 cm has a capacity of V_{1}= 0.5 l of water. Find the dimensions (D_{2} and h_{2}) of a second cup, which has a similar geometry to that of the first cup, if the second cup has a capacity of V_{2}=4 liters of water?
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D2 =16 cm, h2 =20 cm.

Question 10 of 10
10. Question
A vehicle mass is equal to M_{1}=1,050 kg. A similar model car is made with a linear scale of 1 to 60. Determine the mass of the vehicle model M_{2} if it is made from the same material as the car itself.
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