Exercise Volume and surface area
1.
Mrs. Sheila has 1 full can of rice. The can is shaped like a cylinder with a diameter of 28 cm
and a height of 60 cm. Every day, Mrs. Audrey cooks rice by taking 2 cups of rice. Each cup
is shaped like a cylinder with a diameter of 14 cm and a height of 8 cm. How many days will
the rice supply last?
2.
Observe the following image.
A pyramid T.ABCD has a height TP = 8 cm and a slant height
TQ = 10 cm. If the base is a square, then the surface area of the
pyramid is...
3.
A right circular cone is embedded in a sphere such that the rim of the base and the vertex of
the cone are on the surface of the sphere with centre O as shown in the diagram. Given that
the cone has a height of 8 cm and a base radius of 4 cm, find
(a) the radius of the sphere,
(b) the ratio of the volume of the cone to the volume of the sphere.
4.
A cone of base radius 6 cm and height 18 cm is filled with water to a depth of h cm.
The radius of the water surface is 3 cm. The triangles ΔVAB and ΔVOC are similar.
(a) State the value of h.
(b) Find the volume of water that has to be added
to fill the cone completely to the brim.
5.
A solid is in the shape of a cone, with vertical height 7.5 cm and slant height 9 cm.
i.
Find the base radius of the solid.
At 2.56 p.m., the solid is heated and melts at a rate of 4.2 cm³/min.
ii.
Find the time at which the solid has completely melted.
The melted solid is placed into a conical mould with base diameter 8 cm.
iii.
Find the curved surface area of the mould.
6.
A closed inverted conical container with radius 12 cm and height 30 cm contains water to a
depth of 20 cm. The container is inverted. Using similar triangles, find the depth, h cm, of the
water now.
7.
A solid hemisphere has a radius of 3p cm and a total surface area of 912 cm².
(i) Find the value of p. 24 of these hemispheres are melted to form a single hemisphere.
(ii) Carl thinks that the radius of the larger hemisphere is (24×3p) cm. Show that Carl is wrong,
and calculate the correct radius of the larger hemisphere.
8.
The diagram shows a solid formed by removing an identical cone from each end of a cylinder.
The cylinder and the cones have a common radius of 10 cm. The
height of each cone is 10 cm and the height of the cylinder is 24
cm.
(a) Find the volume of the solid.
(b) The solid is made from a material with density 1.15 g/cm³.
Calculate the mass of the solid in kilograms.
9.
A container is made up of a cylinder and a hemisphere. The
container has a base radius of 7 cm and a height of 15 cm.
1000 cm³ of water are poured into the container. Find the
depth of the water in the container.
10. The diagram shows a metal cone of base radius 8 cm and height 15 cm, and a metal
hemisphere of radius 8 cm.
(a) Find the total surface area of
(i) the cone,
(ii) the hemisphere. Give your answers in terms of π.
(b) A solid is formed by joining the plane faces of the cone and the hemisphere together.
Find the percentage decrease in the total surface area of the solids due to the joining.
(c) If the cone and the hemisphere are melted and recast into a cylinder of height 10 cm,
find the base radius of the cylinder.
11. A right prism has a rhombus-shaped base with diagonal lengths of 12 cm and 16 cm. If the
total surface area of the prism is 392 cm², the volume of the prism is...
12. A swimming pool filled with water has dimensions of 20 m in length and 6 m in width. The
water depth at the shallow end is 1 m and gradually deepens to 4 m at the deepest end.
What is the volume of water in the pool in liters?
13. A building roof is shaped like a square pyramid. The length of the base side of the roof is 12
meters and the height of the roof is 8 meters. The surface of the roof will be repaired at a cost
of Rp150,000.00 per square meter. What is the total cost to repair the entire roof?