Cylinders questions with answers

Questions

1. Question 1Volume, in terms of π

   The diagram shows a cylindrical pencil holder. Find its volume, leaving your answer in terms of $\pi$.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{10}{2}$$= 5$ cm
   2. Step 2: Volume of a cylinder = πr²h$V = \pi r^2 h$$= \pi(5)^2(5)$$= 125\pi$ cm³
2. Question 2Volume to 3 s.f.

   The diagram shows a cylinder. Find its volume, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{3.6}{2}$$= 1.8$ m
   2. Step 2: Volume of a cylinder = πr²h: round at the end$V = \pi r^2 h$$= \pi(1.8)^2(6)$$= 61.0725\ldots$$= 61.1$ m³ (3 s.f.)
3. Question 3Curved surface area

   The diagram shows a cylindrical candle. Find its curved surface area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{16}{2}$$= 8$ cm
   2. Step 2: Curved surface area = 2πrh (no ends)$\text{CSA} = 2\pi r h$$= 2\pi(8)(15)$$= 753.9822\ldots$$= 754$ cm² (3 s.f.)
4. Question 4Total surface area, closed or open

   The diagram shows a cylindrical pencil holder, open at the top. Find its total surface area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Curved surface + the circular end(s) it has$\text{TSA} = 2\pi rh + \pi r^2$$= 2\pi(3)(8) + \pi(3)^2$$= 179.0707\ldots$$= 179$ cm² (3 s.f.)
5. Question 5The height from the volume

   The volume of a cylindrical water tank is $2500$ m³. Find its height, $h$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Volume = πr²h$\pi(11)^2 h = 2500$
   2. Step 2: Divide by πr², then round$h = \dfrac{2500}{\pi(11)^2}$$= 6.5766\ldots$$= 6.58$ m (3 s.f.)
6. Question 6Hollow pipes

   The diagram shows a hollow metal pipe of length $15$ cm. Its outer radius is $6$ cm and its inner radius is $4.5$ cm. Find the volume of metal in the pipe, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Volume of metal = outer cylinder − inner (hollow) cylinder$V = \pi(6)^2(15) - \pi(4.5)^2(15)$$= 742.2012\ldots$$= 742$ cm³ (3 s.f.)
7. Question 7The height from the total surface area

   A closed cylinder has a radius of $7$ m and a total surface area of $140\pi$ m². Find its height, $h$ m.

   Show the worked solution

   1. Step 1: Total surface area = two circular ends + the curved surface$2\pi r^2 + 2\pi rh = 140\pi$$2\pi(7)^2 + 2\pi(7)h = 140\pi$
   2. Step 2: Divide by π, then solve for h$98 + 14h = 140$$14h = 42$$h = \dfrac{42}{14}$$h = 3$The height is $3$ m.
8. Question 8Air space round cans in a box

   $8$ identical cylindrical candles, each of radius $2$ cm and height $8$ cm, stand upright in a closed rectangular box: $4$ along its length and $2$ across its width. They fit exactly, touching each other, the sides, the top and the bottom of the box. Find the volume of air space in the box, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The box: each can takes up a diameter (2r) of the length and of the widthLength $= 4 \times 4$$= 16$ cm, width $= 2 \times 4$$= 8$ cm$\text{Box} = 16 \times 8 \times 8$$\phantom{\text{Box}} = 1024$ cm³
   2. Step 2: The 8 cans: 8 × πr²h$\text{Cans} = 8 \times \pi(2)^2(8)$$= 804.2477\ldots$ cm³
   3. Step 3: Air space = box − cans$\text{Air} = 1024 - 8 \times \pi(2)^2(8)$$= 219.7522\ldots$$= 220$ cm³ (3 s.f.)
9. Question 9A rectangle rolled into a cylinder

   A rectangular sheet of paper, $24$ cm by $16$ cm, is rolled up so that its two edges of length $16$ cm just meet, without overlapping. It forms the curved surface of a cylinder of height $16$ cm. Find the radius of the cylinder, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The 24 cm edge goes round the cylinder: it is the circumference$2\pi r = 24$$r = \dfrac{24}{2\pi}$$= 3.8197\ldots$$= 3.82$ cm (3 s.f.)