Composite solids questions with answers

Questions

1. Question 1Volume: cone on a hemisphere

   The diagram shows a solid made of a cone on top of a hemisphere. The cone has height $12$ cm and the common radius is $2.5$ cm. Find the volume of the solid, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Volume of the cone = ⅓πr²hVolume of cone $= \dfrac{1}{3} \times \pi \times 2.5^2 \times 12$$= 78.5398\ldots$ cm³
   2. Step 2: Volume of the hemisphere = ⅔πr³Volume of hemisphere $= \dfrac{2}{3} \times \pi \times 2.5^3$$= 32.7249\ldots$ cm³
   3. Step 3: Volumes add: the solid is both parts togetherTotal volume $= 78.5398\ldots + 32.7249\ldots$$= 111.2647\ldots$$= 111$ cm³ (3 s.f.)
2. Question 2Volume: cylinder with a hemisphere

   The diagram shows a bullet-shaped solid: a cylinder with a hemisphere on top. The cylinder has height $12$ cm and the common radius is $4$ cm. Find the volume of the solid, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Volume of the cylinder = πr²hVolume of cylinder $= \pi \times 4^2 \times 12$$= 603.1857\ldots$ cm³
   2. Step 2: Volume of the hemisphere = ⅔πr³Volume of hemisphere $= \dfrac{2}{3} \times \pi \times 4^3$$= 134.0412\ldots$ cm³
   3. Step 3: Volumes add: the solid is both parts togetherTotal volume $= 603.1857\ldots + 134.0412\ldots$$= 737.2270\ldots$$= 737$ cm³ (3 s.f.)
3. Question 3Surface area: cone on a hemisphere

   The diagram shows a solid made of a cone on top of a hemisphere. The cone has slant height $7$ cm and the common radius is $2.5$ cm. Find the total surface area of the solid, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Which surfaces are on the outside of the solid?The circle where the cone meets the hemisphere is hidden inside, so it is not surface.Surface area $=$ curved surface of cone $+$ curved surface of hemisphere
   2. Step 2: Curved surface of the cone = πrlCurved surface of cone $= \pi \times 2.5 \times 7$$= 54.9778\ldots$ cm²
   3. Step 3: Curved surface of the hemisphere = 2πr²Curved surface of hemisphere $= 2 \times \pi \times 2.5^2$$= 39.2699\ldots$ cm²
   4. Step 4: Add the two curved surfacesTotal surface area $= 54.9778\ldots + 39.2699\ldots$$= 94.2477\ldots$$= 94.2$ cm² (3 s.f.)
4. Question 4Surface area: cylinder with a hemisphere

   The diagram shows a bullet-shaped solid: a cylinder with a hemisphere on top. The cylinder has height $12$ cm and the common radius is $3$ cm. The solid stands on its flat circular base. Find the total surface area of the solid, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Which surfaces are on the outside of the solid?The circle where the cylinder meets the hemisphere is hidden inside, so it is not surface; the flat base is.Surface area $=$ base $+$ curved side of cylinder $+$ curved surface of hemisphere
   2. Step 2: Base circle = πr²Base $= \pi \times 3^2$$= 28.2743\ldots$ cm²
   3. Step 3: Curved side of the cylinder = 2πrhCurved side $= 2 \times \pi \times 3 \times 12$$= 226.1946\ldots$ cm²
   4. Step 4: Curved surface of the hemisphere = 2πr²Hemisphere $= 2 \times \pi \times 3^2$$= 56.5486\ldots$ cm²
   5. Step 5: Add the three partsTotal surface area $= 28.2743\ldots + 226.1946\ldots + 56.5486\ldots$$= 311.0176\ldots$$= 311$ cm² (3 s.f.)
5. Question 5Solids in containers: empty space, rising water

   A solid cone of base radius $4$ cm and height $10$ cm stands inside a cylindrical container of base radius $4$ cm. Water is poured in until it is $12$ cm deep. Find the volume of water in the container, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The space up to a depth of 12 cm: a cylinderCylinder to depth $12$ cm $= \pi \times 4^2 \times 12$$= 603.1857\ldots$ cm³
   2. Step 2: The cone takes up some of that space (it is under the water)Cone $= \dfrac{1}{3} \times \pi \times 4^2 \times 10$$= 167.5516\ldots$ cm³
   3. Step 3: Water = the space − the coneWater $= 603.1857\ldots - 167.5516\ldots$$= 435.6341\ldots$$= 436$ cm³ (3 s.f.)
6. Question 6A cone with its top cut off (frustum)

   The diagram shows a solid frustum (a cone with its top cut off): a large cone with a small cone cut off its top. The circular faces have radii $4.5$ cm and $3$ cm, and the frustum is $3$ cm high. Find the volume of the frustum, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Similar triangles: the small cone's height over its radius is the large cone's height over its radius$\dfrac{h}{3} = \dfrac{h + 3}{4.5}$$4.5h = 3h + 9$$h = 6$ cm, so the large cone is $9$ cm high.
   2. Step 2: Volume of the large coneLarge cone $= \dfrac{1}{3} \times \pi \times 4.5^2 \times 9$$= 190.8517\ldots$ cm³
   3. Step 3: Volume of the small coneSmall cone $= \dfrac{1}{3} \times \pi \times 3^2 \times 6$$= 56.5486\ldots$ cm³
   4. Step 4: Frustum = large cone − small coneFrustum $= 190.8517\ldots - 56.5486\ldots$$= 134.3030\ldots$$= 134$ cm³ (3 s.f.)
7. Question 7A cone on a wider cylinder: the ring left showing

   The diagram shows a solid made of a cylinder of radius $5$ cm and height $5$ cm, with a cone of base radius $4$ cm and height $3$ cm standing on the centre of its top face. Find the total surface area of the solid, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The cone's slant height: l² = r² + h²$l = \sqrt{4^2 + 3^2}$$\phantom{l} = 5$
   2. Step 2: The surfaces you can touch: the base, the curved side, the ring left on top, and the cone's curved surfaceBase: $\pi \times 5^2 = 25\pi$Curved side: $2\pi \times 5 \times 5 = 50\pi$Ring: $\pi \times 5^2 - \pi \times 4^2 = 9\pi$Cone: $\pi \times 4 \times 5 = 20\pi$
   3. Step 3: Add themTotal $= 104\pi$$= 326.7256\ldots$$= 327$ cm² (3 s.f.)
8. Question 8Surface area in terms of π and r

   A solid is made of a hemisphere, a cylinder and a cone, each of radius $r$ cm. The cylinder has height $r$ cm, the cone has height $\dfrac{5}{12}r$ cm, and the hemisphere and the cone are joined to the two ends of the cylinder. Find the total surface area of the solid in the form $k\pi r^2$ cm², where $k$ is a number.

   Show the worked solution

   1. Step 1: The cone's slant height: l² = r² + h²$l^2 = r^2 + \left(\dfrac{5}{12}r\right)^2$$\phantom{l^2} = \dfrac{169}{144}r^2$$l = \dfrac{13}{12}r$
   2. Step 2: The three curved surfaces: hemisphere 2πr², cylinder 2πr × height, cone πrl$2\pi r^2 + 2\pi r \times r + \pi r \times \dfrac{13}{12}r$$= 2\pi r^2 + 2\pi r^2 + \dfrac{13}{12}\pi r^2$
   3. Step 3: Add them (the flat circles where the parts join are hidden)$k = \dfrac{61}{12}$
9. Question 9A pyramid on a cuboid

   The diagram shows a solid made of a cuboid with a square base of side $8$ cm and height $4$ cm, and a right pyramid of height $3$ cm on its top face. Find the total surface area of the solid.

   Show the worked solution

   1. Step 1: Slant height of a triangular face: the height and half the side$l^2 = 3^2 + 4^2$$l = 5$
   2. Step 2: The faces you can touch: the bottom square, the four sides of the cuboid and the four triangles (the top of the cuboid is hidden)Bottom: $8^2 = 64$Four sides: $4 \times 8 \times 4 = 128$Four triangles: $4 \times \dfrac{1}{2} \times 8 \times 5 = 80$
   3. Step 3: Add them$= 272$ cm²