Spheres questions with answers

Questions

1. Question 1Volume of a sphere

   The diagram shows a globe of diameter $9$ cm. Find its volume, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{9}{2}$$= 4.5$
   2. Step 2: Volume of a sphere = ⁴⁄₃πr³$V = \dfrac{4}{3}\pi r^3$$V = \dfrac{4}{3} \times \pi \times 4.5^3$$= 381.7035\ldots$$= 382$ cm³ (3 s.f.)
2. Question 2Surface area of a sphere

   The diagram shows a ball of diameter $8$ cm. Find its surface area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{8}{2}$$= 4$
   2. Step 2: Surface area of a sphere = 4πr²Surface area $= 4\pi r^2$Surface area $= 4 \times \pi \times 4^2$$= 64\pi$$= 201.0619\ldots$$= 201$ cm² (3 s.f.)
3. Question 3Hemispheres

   The diagram shows a solid hemisphere of diameter $9$ cm. Find its volume, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{9}{2}$$= 4.5$
   2. Step 2: A hemisphere is half a sphere: volume = ½ × ⁴⁄₃πr³ = ⅔πr³$V = \dfrac{2}{3}\pi r^3$$V = \dfrac{2}{3} \times \pi \times 4.5^3$$= 190.8517\ldots$$= 191$ cm³ (3 s.f.)
4. Question 4The radius from the volume or surface area

   The volume of a solid sphere is $972\pi$ cm³. Find its radius, $r$ cm.

   Show the worked solution

   1. Step 1: Volume = ⁴⁄₃πr³: write an equation in r$\dfrac{4}{3}\pi r^3 = 972\pi$$r^3 = 972 \times \dfrac{3}{4}$$= 729$
   2. Step 2: Cube-root to find r$r = \sqrt[3]{729}$$= 9$ cm
5. Question 5Melting and recasting

   A solid metal cylinder of radius $5$ cm and height $12$ cm is melted down and recast into small solid spheres, each of radius $0.8$ cm. Find the greatest number of spheres that can be made.

   Show the worked solution

   1. Step 1: The volume of metal (melting doesn't change it)Volume of the cylinder $= \pi \times 5^2 \times 12$$= 942.4777\ldots$ cm³
   2. Step 2: Volume of one small sphere = ⁴⁄₃πr³Volume of one sphere $= \dfrac{4}{3} \times \pi \times 0.8^3$$= 2.1446\ldots$ cm³
   3. Step 3: Divide, then round DOWN: only whole spheres can be made$\dfrac{942.4777\ldots}{2.1446\ldots} = 439.4531\ldots$The greatest number of spheres is $439$.
6. Question 6Coating and painting: cost, tins

   A factory makes $120$ solid balls, each of radius $2.5$ cm. The whole surface of each ball is coated with paint costing $0.35 for every $100$ cm². Find the total cost of the coating, correct to the nearest cent.

   Show the worked solution

   1. Step 1: Area of one ball: 4πr²Area of one $= 4 \times \pi \times 2.5^2$$= 78.5398\ldots$ cm²
   2. Step 2: All 120 of themTotal area $= 120 \times 78.5398\ldots$$= 9424.7779\ldots$ cm²
   3. Step 3: Cost = number of 100 cm² × the price for each 100 cm², to the nearest cent$\dfrac{9424.7779\ldots}{100} \times 0.35 = 32.9867\ldots$The cost is $32.99.
7. Question 7Recast into a prism with a sector cross-section

   A solid metal sphere of radius $3$ cm is melted down and recast into a prism of length $15$ cm, with no metal lost. The cross-section of the prism is a sector of a circle with radius $R$ cm and angle $90^\circ$, as shown. Find $R$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Volume of the metal (keep it in terms of π)$V = \dfrac{4}{3}\pi \times 3^3$$\phantom{V} = 36\pi$ cm³
   2. Step 2: Volume of the prism = area of the sector × length, with the sector's area = (θ/360) × πR²$\dfrac{90}{360} \times \pi R^2 \times 15 = 36\pi$
   3. Step 3: The π cancels: make R² the subject$R^2 = \dfrac{36 \times 360}{90 \times 15}$$\phantom{R^2} = 9.6$
   4. Step 4: Take the square root$= 3.0983\ldots$$= 3.10$ cm (3 s.f.)