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IB Mathematics: Applications and Interpretation SL · 12.10: Geometry & Trigonometry · 12.10.9

Name Date

Volumes and surface areas of solids Problem set B

Sketch a cross-section when the solid is made of more than one part. Give answers to 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A cone has base radius 4.6 cm and perpendicular height 8.3 cm.

    8.3 cm4.6 cm
    1. (a) Find the slant height.
    2. (b) Find the volume.

  2. A cone has base radius 5.5 cm and slant height 14.2 cm.

    1. (a) Find the curved surface area.
    2. (b) Find the total surface area, including the base.

  3. A sphere has radius 6.4 cm.

    1. (a) Find its volume.
    2. (b) Find its surface area.
    3. (c) A solid hemisphere has the same radius. Find its total surface area, including the flat face.

  4. A sphere has volume 900 cm³.

    1. (a) Find the radius of the sphere.
    2. (b) A cone has volume 450 cm³ and base radius 6 cm. Find its perpendicular height.

In context

  1. A hemispherical fish tank has internal diameter 42 cm.

    1. (a) Find the capacity in litres.
    2. (b) How many 0.5 litre bottles can be filled completely from the full tank?

  2. A drinks flask is a cylinder of radius 4.5 cm and height 20 cm, closed by a hemispherical cap of the same radius.

    9 cm20 cm
    1. (a) Find the volume.
    2. (b) Find the area of its curved outside surface (wall and cap, not the base).

  3. A tent is a cuboid with a square base of side 3.6 m and height 1.6 m, with a square-based pyramid roof of height 1.2 m, with the same base.

    3.6 m1.6 m
    1. (a) Find the total volume.
    2. (b) Find the area of canvas needed for the four sloping triangular faces of the roof.

  4. A solid steel ball of radius 5.4 cm is melted and made into 9 identical cones, each with base radius 2.7 cm. No steel is lost. Find the perpendicular height of each cone.

Stretch

  1. A pencil-top ornament is a solid hemisphere of radius 4 cm joined to the base of a cone of the same base radius. The volume of the whole ornament is 300 cm³. Find the perpendicular height of the cone.

  2. A conical paper cup has internal radius 9 cm at the top and internal depth 12 cm. It stands on its point and holds juice to a depth of 9 cm.

    18 cm9 cm
    1. (a) Find the radius of the liquid surface.
    2. (b) Find the volume of liquid.
    3. (c) Find the volume of liquid as a percentage of the volume of the full container.

jmaths.xyz · AI SL · 12.10.9 Volumes and surface areas of solids