Volumes and surface areas of solids Problem set B Worked solutions

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

1. (a) Find the slant height.

   • \(l=\sqrt{4.6^2+8.3^2}=9.489\) cm
   • 9.49 cm

2. (b) Find the volume.

   • \(V=\frac13\pi r^2h=\frac13\pi(4.6)^2(8.3)=183.92\)
   • 184 cm³

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

1. (a) Find the curved surface area.

   • \(\pi rl=\pi(5.5)(14.2)=245.36\)
   • 245 cm²

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

   • \(\pi r^2=95.03\), so total \(=245.36+95.03=340.39\)
   • 340 cm²

A sphere has radius 6.4 cm.

1. (a) Find its volume.

   • \(\frac43\pi(6.4)^3=1098.07\)
   • 1100 cm³

2. (b) Find its surface area.

   • \(4\pi(6.4)^2=514.72\)
   • 515 cm²

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

   • Curved \(2\pi r^2=257.36\), flat face \(\pi r^2=128.68\)
   • Total \(3\pi r^2=386.04\), so 386 cm²

A sphere has volume 900 cm³.

1. (a) Find the radius of the sphere.

   • \(\frac43\pi r^3=900\)
   • Solve for \(r\): \(r=5.989\)
   • 5.99 cm

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

   • \(\frac13\pi(6)^2h=450\)
   • \(h=\frac{3\times450}{\pi\times36}=11.937\)
   • 11.9 cm

A hemispherical fish tank has internal diameter 42 cm.

1. (a) Find the capacity in litres.

   • \(r=21\) cm, \(V=\frac23\pi(21)^3=19396.2\) cm³
   • \(19396.2\div1000=19.396\) litres, so 19.4 litres

2. (b) How many 0.5 litre bottles can be filled completely from the full tank?

   • \(19.396\div0.5=38.79\)
   • 38 complete bottles

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

1. (a) Find the volume.

   • Cylinder \(\pi(4.5)^2(20)=1272.35\), hemisphere \(\frac23\pi(4.5)^3=190.85\)
   • Total 1463.2

2. (b) Find the area of its curved outside surface (wall and cap, not the base).

   • Wall \(2\pi(4.5)(20)=565.49\), curved top \(2\pi(4.5)^2=127.23\)
   • Total 692.7

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.

1. (a) Find the total volume.

   • Cuboid \(3.6\times3.6\times1.6=20.74\), pyramid \(\frac13(3.6)^2(1.2)=5.184\)
   • Total 25.92

2. (b) Find the area of canvas needed for the four sloping triangular faces of the roof.

   • Slant height \(\sqrt{1.2^2+1.8^2}=2.163\)
   • Four faces: \(4\times\frac12\times3.6\times2.163=15.58\)

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.

• Sphere: \(\frac43\pi(5.4)^3=659.58\) cm³
• 9 cones: \(9\times\frac13\pi(2.7)^2h=68.707\,h\)
• \(h=\frac{659.58}{68.707}=9.600\), so 9.60 cm

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.

• Hemisphere: \(\frac23\pi(4)^3=134.04\)
• Cone: \(\frac13\pi(4)^2h\). Total: \(134.04+\frac13\pi(4)^2h=300\)
• \(h=\frac{3(300-134.04)}{\pi(4)^2}=9.905\), so 9.90 cm

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.

1. (a) Find the radius of the liquid surface.

   • \(\frac{r}{9}=\frac{9}{12}\)
   • \(r=6.750\), so 6.75 cm

2. (b) Find the volume of liquid.

   • \(\frac13\pi(6.750)^2(9)=429.42\)
   • 429 cm³

3. (c) Find the volume of liquid as a percentage of the volume of the full container.

   • Full container \(\frac13\pi(9)^2(12)=1017.9\)
   • \(\frac{429.42}{1017.9}\times100=42.2\)%