Part I - Geometry

University of Cebu College of Engineering Engineering Mathematics

1. A thin sheet of metal in the form of a sector of a circle is 72 cm in diameter with a central angle of 270°. Compute the volume of a cone. ANS. 18178 cu.cm 2. The lateral height of a right circular cone is 40π sq.m. The base radius is 4 m. What is the slant height? ANS. 10 m 3. A cardboard in the form of a sector of a circle is formed into a cone. If the sector has a radius of 36 cm and a central angle of 90°, what is the volume of a cone in cu.cm? ANS. 2957 cu.cm 4. The slant height of a right circular cone is 5 m. What is the volume of the cone in cu.m if the base diameter is 6 m? ANS. 37.70 m³ 5. A right circular cone with vertical axis and with base uppermost is surmounted by a hemisphere. If the volume of the hemisphere is twice that of the cone, what angle is formed by a slant height and the axis of the cone? ANS. ∅ = 45° 6. A cone with a volume of 2800 cu.m has a horizontal base having a diameter of 22 cm. The axis makes an angle of 56° with the base. What is the length of the axis in cm? ANS. 26.7 cm 7. A cone has an axis of 30 cm inclined at 60° with the horizontal base. The horizontal base has a diameter of 20 cm. What is the volume of the cone? ANS. 2720.6 cu.cm 8. The axis of the cone, 2 m long makes an angle of 65° with it’s horizontal base. What is the area of the base if it’s volume is 5 cu.m. ANS. 8.3 m² 9. The 30 cm axis of a cone is inclined at 55° with the horizontal base. If it’s volume is 2500 cu.m what is the diameter of it’s horizontal circular base? ANS. 19.7 cm 10. A plane parallel to the base of a right circular cone cuts the cone x (meters) from the vertex. If the volume of the smaller cone is ½ times the volume of the big cone and the height of the big cone is 10 m. ANS. 7.94 m 11. What is the lateral area of the frustrum of a right circular cone whose base diameters are 2 m and 4 m if it’s altitude is 5 m. ANS. 48.06 m² 12. The frustrum of a right circular cone has an upper base radius of 2 m and lower base radius of 1 m. If the distance between the bases is 4 m, what is the volume of the frustrum of the cone? ANS. 29.32 m³ 13. The frustrum of a right circular cone has an altitude of 7 m and a volume of 51.31 cu.m.

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If the upper base diameter is 2 m, what is the lower base diameter? ANS. 4 m The upper and lower base radii of the frustrum of a right circular cone are 2 m and 1 m respectively. If the volume is 43.98 m, what is the altitude in meters? ANS. 6 m The volume of the frustrum of a right circular cone is 36.65 cu.m. The distance between the bases is 5 m. If the upper base radius is 2 m, what is the upper base radius? ANS. 1 m The center of gravity of a regular triangular pyramid is 3.675 m from the vertex of the pyramid. Determine the side of the triangle face. ANS. 6 m A regular pyramid having a volume of 800 m³ has a horizontal base in the form of an equilateral triangle. If the altitude is 12 m, calculate for the lateral area. ANS. 435.7 m² A regular hexagonal pyramid has a volume of 280.59 cu.m and an altitude of 9 m. What is the base edge in meters? ANS. 6 m A regular square pyramid has an altitude of 20 cm and a lateral area of 1500 sq.cm. What is the base edge in cm? ANS. 30 cm Find the surface area of a regular pyramid whose volume is 25.46 m³. ANS. 62.4 m A pyramid has an altitude of 20 cm and a square base of 8 cm by 8 cm. It is cut parallel to the base at a distance of 7 cm from the base. Determine the volume of the smaller pyramid. ANS. 117 cm³ The upper base of a frustrum of a regular triangular pyramid is an equilateral triangle having an area of 3.897 sq.m. The volume of the frustrum is 67.55 cu.m. If the bases are 4 m apart, what is the lower base edge? ANS. 9 m The volume of the frustrum of a regular triangular pyramid is 135 cu.m. The lower base is an equilateral triangle with an edge of 9 m. The upper base is 8 m above the lower base. What is the upper base edge? ANS. 3 m The upper and lower bases of the frustrum of a regular rectangular pyramid are 3 m by 4 m and 6 m by 8 m respectively. What is the distance between the bases? ANS. 5 m The bases of a frustrum of a pyramid are 18 cm by 18 cm and 10 cm by 10 cm. It’s lateral area is 448 cm². What is the altitude of the frustrum? ANS. 6.93 cm The upper and lower bases of a regular pyramid are 4 m by 4 m and 6 m by 6 m. Each lateral edge is 2.5 cm long. Determine: 26.1. Volume of the frustrum

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26.2. Lateral area of the frustrum 26.3. Total surface area ANS. 52.2 m³, 45.8 m², 97.8 m² The volume of the frustrum of a regular rectangular pyramid is 93.333 cu.m. The upper base is 2.5 m by 4 m. What is the area of the lower base? ANS. 40 m² The lower base of the frustrum of a regular rectangular pyramid is 5 m by 8 m. The volume is 116.667 cu.m and the altitude is 5 m. What is the area of the upper base? ANS. 10 m² The frustrum of a square pyramid has a lower base 24 cm x 24 cm and an upper base of 15 cm x 15 cm. If the altitude is 10 cm, what is the lateral area? ANS. 855.35 cm² A regular square pyramid has a horizontal base 26 cm by 26 cm and an altitude of 50 cm. A plane is passed parallel to the base and 28 cm from the vertex. What is the lateral area of the part of the pyramid below the plane? ANS. 1843.8 cu.cm If each edge of a cube is increased by 50%, by how much, in percentage, will the volume increase? ANS. 237.5% If the edge of the cube is increased by 30%, by how much will the surface area increase? ANS. 69% A regular hexahedron has an edge of 32 cm long. What is the ratio of it’s total surface area to it’s volume? ANS. 0.1875 If there is a block of wood in the form of rectangular parallelipiped 2’ by 3’ by 4’ whose faces are painted red. 34.1. How many times must you cut completely through the whole block to form cubes 1’ by 1’ by 1’? 34.2. How many cubes will there be? 34.3. How many cubes will have 3 red faces? 34.4. How many cubes will have two red faces? 34.5. How many cubes will have one red face? ANS. 6 cuts, 24, 8, 12, 4 cubes The sphere of a cubical box touch the spherical she’ll that encloses it. If the volume of the box is 27000 cu.cm: 35.1. What is the volume of the sphere? 35.2. What is the volume of the space outside box but inside the sphere? ANS. 73459 cu.cm, 46459 cu.cm A circular cylindrical tank having a base diameter of one meter and a length of 2 m is partly filled with water. When it’s axis is horizontal, the water is 2/3 of the diameter deep. How deep will the water be when its axis is vertical? ANS. 1.42 m 68 sq.m of sheet metal was used to construct a cylindrical tank with open top.

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The tanks diameter is 2/3 of its height. Find the height of the tank. ANS. 5.28 m The lateral surface of a circular column is covered with 5.65 sq.m of wall paper. If the column is 3 m high, find its diameter. ANS. 60 cm A cylindrical tank can hold 54 cu.m of oil. The tanks height is 1.5 times it’s diameter. Find its height. ANS. 5.37 m 500 cu.m of water are to be stored in eight cylindrical tanks with a diameter of 4 m. What should be the height of each tank? ANS. 4.97 m Find the area of the heptagon inscribed in a circle if the area of the circle is 210 sq.m. ANS. 182.9 m² Find the area of a pentagon that is inscribed in a circle if the area of the circle is 314 m². ANS. 237.6 m² Find the area of a decagon that is inscribed in a circle if the area of the circle is 412 m². ANS. 385.4 m² Find the area of a hexagon that is inscribed in a circle if the area of the circle is 170 m². ANS. 140.6 m² Find the area of nonagon that is inscribed in a circle if the area of the circle is 94 m². ANS. 86.5 m² A circle with an area of 300 m² is inscribed in a pentagon. Find the area of the pentagon. ANS. 346. 9 m² The area of a circle that is inscribed in a heptagon is 128 m. What is the area of the heptagon. ANS. 137.3 m² A circle is circumscribed about a hexagon. The difference between the area of the two is 96 m². Find the area of the hexagon. ANS. 458.9 m² A circle is circumscribed about a hexagon. The area outside the hexagon but inside the circle is 15 m². Find the area of the hexagon. ANS. 71.70 m² Find the area of a pentagram inscribed in a circle having a radius of 5 cm. ANS. 45.41 sq.cm The volume of a right prism is 500 cu.cm. The bases are hexagons with each side 6 cm long. What is the distance between the bases in cm? ANS. 5.35 cm A right prism has hexagonal bases with an edge of 6 cm. The bases are 12 cm apart. What is the volume of the prism? ANS. 1122.4 cm³ A right prism has pentagonal base ls which are 9 cm apart. If it’s volume is 557.415 cu.cm, what is its total surface area? ANS. 393.84 cm² The altitude of a right prism is 15 cm and it’s volume is 929.025 cm³. What is the lateral area if the bases are pentagons? ANS. 450 cm

55. The volume of a right prism whose height is 6 cm is 420 cu.cm. Each base has 6 equal sides. What is the base edge? ANS. 5.19 cm 56. Three identical circles are tangent to one another. The area enclosed by three circles but outside each of the three circles is 16.13 sq.m. What is the area of each circle? ANS. 314.2 sq.cm 57. A circle having an area of 452 m is cut into two segments by a chord which is 6 m from the center of the circle. Compute the area of the biggest segment. ANS. 363.68 m² 58. Two chords of a circle intersect and divide each other into two line segments. The segments of the first chord are 6 cm and 3 cm long. One segment of the second chord is 2 cm long. How long is the remaining segment? ANS. 9 cm 59. A road runs tangential to a circular lake 8 km long from the point of tangency. Along the road, a new road and bridge were constructed to cross the lake. This new road is extended to cross the lake. If the bridge is 2.14 km long, how long is the new road before the bridge? ANS. 7 km 60. Three identical circles are tangent to one another. The radius is 20 cm. Find the area enclosed by the three circles but outside each of the three circles in sq.cm. ANS. 64.5 sq.cm 61. A circle with a diameter of 8 cm is inscribed in a circular sector with a central angle of 80°. What is the area of the sector? ANS. 72.96 cm² 62. QRS is a circular sector with a central angle of 40° at 0 and a radius of 20 cm. RS is the arc. From point R, a line is drawn to point C, the mid point of QS. Find the area of RCS in sq.cm. ANS. 75.3 sq.cm 63. The area of a rhombus is 132 sq.cm. It has one diagonal equal to 12 cm. Determine: 63.1. The sides of the rhombus 63.2. The acute angle between the sides of a rhombus ANS. 22, 12.53, 57.22° 64. Two sides of a parallelogram are 68 cm and 83 cm and one of it’s diagonals is 42 cm. Solve for the biggest interior angle of the parallelogram. ANS. 149.73° 65. Find the area of a quadrilateral having sides AB = 10 m, BC = 5 m, CD = 14.14 m, and DA = 15 m if the sum of opposite angles is equal to 225°. ANS. 100 m 66. Two sides of a parallelogram are 68 cm and 83 cm. What is the difference in lengths of the diagonals? ANS. 103.81 cm 67. The sides of a cyclic quadrilateral are a = 3 m, b = 3 m, c = 4 m and d = 4 m. Find the radius of the circle that can be inscribed in it. ANS. 1.71 m 68. A truncated prism has a horizontal triangular base ABC. AB = 30 cm, BC = 40 cm, and

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CA = 50 cm. The vertical edges through A, B, and C are 18 cm. Compute the volume of the prism. ANS. 13800 cu.cm A truncated prism has a base which is a square right section. The edges perpendicular to the base are 20 cm, 17 cm, and 20 cm long. Determine the length of a side of the base if the volume of the prism is 5780 cu.cm. ANS. 19 cm The volume of a truncated prism is 6240 cu.cm. The base is a right section in the form of equilateral triangle. The edges perpendicular to the base are 15 cm, 18 cm, and 15 cm long. Find the length of one side of the base. ANS. 30 cm How far from any vertex is the center of gravity of a tetrahedron if an edge is 50 cm. ANS. 30.6 cm What is the altitude of a regular tetrahedron whose surface area is 140 m². ANS. 7.3 m Determine the total surface area of a regular tetrahedron whose volume is 12.3 cu.m ANS. 38.4 sq.m What is the surface area of a regular tetrahedron whose altitude is 12 m? ANS. 374.3 m² How far from the vertex is the opposite side of a tetrahedron if the edge is 50 cm? ANS. 40.82 cm Each face of the octahedron is an equilateral triangle. If it’s vlume is 58917 cu.cm, how long is the side of the triangle? ANS. 50 cm. The number of vertices of a dodecahedron is: ANS. 20 The total surface area of a dodecahedron is 4645.3 sq.cm. If each face is a regular pentagon, find the length of its side. ANS. 15 cm The number of edges of an icosahedron is: ANS. 30 One side of the faces of an icosahedron is anequilateral triangle. If the length of one of the side is 8 cm, compute its volume. ANS. 1116.16 cm² Find the ratio of the surface area of a sphere to it’s volume if the radius is 8 cm. ANS. 3:8 What is the surface area of the sphere whose volume is 36 cu.cm? ANS. 52.7 cm² The volume of a sphere in cu.cm is equal to twice its surface area in sq.cm. Find its radius. ANS. 6 cm A spherical steel ball is placed into a vertical circular cylinder containing water, causing the water level to rise by 12 cm. If the radius of the cylinder is 15 cm, what is the radius of the ball? ANS. 12.65 cm A spherical steel ball having a radius of 10 cm is placed into a circular cylinder containing water. If the radius of the cylinder is 12 cm, by how much will the water level rise? ANS. 9.26 cm

86. A spherical shell has a thickness of 4 cm and a solid volume of 0.33805 cu.m. What is the internal radius? ANS. 0.8 mThe surface area of a sphere in cm² is numerically equal to its volume. Find the volume. ANS. 113.10 cu.cm 87. A spherical shell has a thickness of 2 cm and an external radius of one meter. Determine it’s solid volume. ANS. 0.256 cu.m 88. A spherical tank is partly filled with 22.9 cu.m of liquid. Find the depth of the liquid from the bottom of the tank if the diameter is 400 cm. ANS. 250 cm 89. 5 cu.m of water is inside a spherical tank whose radius is 2 m. Find the height of the water surface above the bottom of the tank. ANS. 2.3 m 90. A sphere has a diameter of 30 cm. The altitude of the first segment is 6 cm. What is the ratio of area of the zone of the 2nd segment to that of the first? ANS. 4:1 91. The radius of a spherical sector with a central angle of 60° is 12 cm. Compute the area of the zone of the spherical sector. ANS. 121.22 cm 92. The level of gasoline in a spherical tank 6 m in diameter dropped from 5.5 m to 3.5 m. How much gasoline was taken out? ANS. 40.3 cu.m 93. One meter curved strip around and above the base of a hemispherical dome is to be painted with two coats of enamel which has a spreading capacity of 200 sq.ft per gallon. Determine the number of gallons of paint needed if the diameter of the dome is 16 m. ANS. 5.39 gallons 94. A spherical segment has 2 parallel bases. The bigger base is a great circle with a radius of 80 cm. The smaller base has a radius of 30 cm. Find the area of the zone. ANS. 37278 sq.cm 95. A spherical sector is cut from a sphere whose radius is 12 cm. Find its volume if it’s central angle is 30°. ANS. 123.3 cu.cm 96. A sphere is whittled down into a spherical sector with a central angle of 53° and a volume of 420 cu.cm. Find the radius of the sphere. ANS. 12.4 cm 97. Find the volume of a spherical wedge having a radius of 2 m and a central angle of 1.25 radians. ANS. 6.7 cu.m 98. A spherical wedge has a curve surface area of 9.2 sq.m and a radius of 2.36 m. Determine it’s volume. ANS. 7.24 cu.m 99. A spherical sector is cut from a sphere such that its central angle is 25° and it’s volume is 185 cu.cm. Determine the radius of the sphere. ANS. 15.50 cm. 100. What is the radius of the spherical wedge whose curve surface area is 12.4 sq.m with a central angle of 1.08 radians? ANS. 2.4 m