Solid Geometry

Question 1

The areas of two similar solids are 24 sq in and 96 sq in. Find the ratio of their volumes.

a) V1/V2 = 1/4
b) V1/V2 = 1/9
c) V1/V2 = 1/8
d) V1/V2 = 1/5

Answer 1

V1/V2 = 1/8

Question 2

A trough whose ends are isosceles trapezoid has a vertical axis of 10m long. The lower base of the trapezoid is 2m, and the upper base is 6m, and is 4m deep. The trough contains 100,000 liters of water.

How deep is the water in the trough?
a) 3.1 m
b) 3.0 m
c) 2.9 m
d) 2.8 m

Determine the wetted area of the trough.
a) 105.81 m2
b) 108.51 m2
c) 108.41 m2
d) 104.81 m2

Answer 2

c) 2.9 m

d) 104.81 m2

Question 3

In a rectangular parallelepiped, the length and width of the base are 12 in and 9 in, respectively. Find the volume of the solid if the length of the diagonal of the solid is 25 inches.

a) 2,610 in2
b) 3,160 in2
c) 2, 610 in2
d) 2,160 in2

Answer 3

d) 2,160 in2

Question 4

A closed right cylindrical tank 8 m high and 4 m in diameter is 1/4 full of water. What is the height of the water when it is lying in horizontal position?

a) 1.16 m
b) 1.17 m
c) 1.18 m
d) 1.19 m

Answer 4

d) 1.19 m

Question 5

A regular hexagonal prism has its axis inclined at 50° from the horizontal. The length of this axis is 6 m, and the apothem of the base is 3 cm. What is its volume?

a) 133.3 m3
b) 143.3 m3
c) 153.3 m3
d) 163.3 m3

Answer 5

b) 143.3 m3

Question 6

Two vertical conical tanks are joined at the vertices by a pipe. Initially, the bigger tank is full of water. The pipe valve is open to allow the water to flow to the smaller tank until it is full. At this moment, how deep is the water in the bigger tank? The bigger tank has a diameter of 6 ft. and a height of 10 ft., while the smaller tank has the same diameter but a height of 8 ft. Neglect the volume of water in the pipeline.

a) 2.85 ft
b) 3.85 ft
c) 4.85 ft
d) 5.85 ft

Answer 6

d) 5.85 ft

Question 7

A regular triangular pyramid has an altitude of 9 m and a volume of 46.8 m3.

Find the length of the edges.
a) 8.61 m
b) 9.17 m
c) 9.92 m
d) 10.42 m

Find the lateral area of the pyramid.
a) 9.47 m
b) 9.53 m
c) 9.64 m
d) 9.71 m

Answer 7

b) 9.17 m

c) 9.64 m

Question 8

An inverted square pyramid has a height equal to 8 m and a top edge equal to 3 m. Initially, it contains water to a depth of 5 m.

What is the initial volume of the water in the tank?
a) 3.86 m3
b) 4.85 m3
c) 5.86 m3
d) 6. 86 m3

What is the wetted area of the pyramid?
a) 17.95 m2
b) 18.09 m2
c) 19.08 m2
d) 20.23 m2

Answer 8

c) 5.86 m3

c) 19.08 m2

Question 9

Find the volume of a right circular cone to be obtained from a sector of radius 26 cm and whose central angle measures 140.5°.

a) 5218.15 cm3
b) 5281.15 cm3
c) 2518.15 cm3
d) 2581.15 cm3

Answer 9

d) 2581.15 cm3

Question 10

The volume of a conical frustum is 2401π cm3. If the radius of the lower base is 14 cm and the height is 21 cm., find the lateral area of the conical frustum.

a) 1640.38 cm3
b) 1460.38 cm3
c) 1640.83 cm3
d) 1460.83 cm3

Answer 10

b) 1460.38

Question 11

The base edges of a frustum of a regular square pyramid are 6 cm and 10 cm. The slant height of the frustum makes with the lower base an angle of 60°. Find the total area of the frustum.

a) 244 cm2
b) 254 cm2
c) 264 cm2
d) 274 cm2

Answer 11

c) 264 cm2

Question 12

A sphere is cut by a plane, if the radius of the plane cut is 4 cm and the diameter of the great circle is 10 cm, determine the:

Volume of the sphere.
a) 523.6 cm3
b) 345.2 cm3
c) 834.1 cm3
d) 413.2 cm3

Volume of the smaller spherical segment.
a) 34.43 cm3
b) 65.56 cm3
c) 54.45 cm3
d) 29.92 cm3

Total surface area of the smaller segment.
a) 93.10 cm2
b) 103.10 cm2
c) 113.10 cm2
d) 123.10 cm2

Answer 12

a) 523.6 cm3

c) 54.45 cm3

c) 113.10 cm2

Question 13

A mixture compound of equal parts of two liquids, one white and the other black, was placed in a hemispherical bowl. The total depth of the two liquids is 6 inches. After standing for short time, the mixture separated, the white liquid settling below the black. If the thickness of the segment of the black liquid is 2 inches:

Find the radius of the bowl in inches.
a) 6.33 in
b) 7.33 in
c) 8.33 in
d) 9.33 in

Find the volume of the bowl.
a) 825.97 in3
b) 967.20 in3
c) 721.13 in3
d) 782.53 in3

Answer 13

b) 7.33 in

a) 825.97 in3

Question 14

The volume of spherical segment with two bases is equal to 159π m3. If the radii of the bases are 4 m and 5 m, respectively, find the:

Distance between the two bases.
a) 2 m
b) 4m
c) 6 m
d) 8 m

Radius of the sphere.
a) 4.58 m
b) 4.85 m
c) 5.84 m
d) 5.48 m

Area of zone of the spherical segment.
a) 196.70 m2
b) 197.60 m2
c) 206.70 m2
d) 207.60 m2

Answer 14

c) 6 m

d) 5.48 m

c) 206.70 m2

Question 15

A spherical sector (spherical cone) has a central angle of 60° and the radius of the sphere is 15 cm.

Find the volume of the spherical sector.
a) 947.01 cm3
b) 974.01 cm3
c) 1047.01 cm3
d) 1074.01 cm3

Determine the total surface area of the spherical sector.
a) 524.83 cm2
b) 542.83 cm2
c) 624.83 cm2
d) 642.83 cm2

Answer 15

a) 947.01 cm3

b) 542.83 cm2

Question 16

A spherical wedge has dihedral angle of 50°. It is from a sphere that has a radius of 10 cm. Find the:

Volume of the wedge.
a) 561.78 cm3
b) 571.78 cm3
c) 581.78 cm3
d) 591.78 cm3

Area of the corresponding lune.
a) 174.52 cm2
b) 184.52 cm2
c) 194.52 cm2
d) 204.52 cm2

Total surface area of the wedge.
a) 188.69 cm2
b) 288.69 cm2
c) 388.69 cm2
d) 488.69 cm2

Answer 16

c) 581.78 cm3

a) 174.52 cm2

d) 488.69 cm2