(a) Calculate the volume of
(i) a solid cylinder with radius 6 cm and height 14 cm.
(ii) a solid hemisphere with radius 6 cm.
[The volume, V, of a sphere with radius r is V = \(\frac{4}{3}\pi r^3\).]
(b) 
The cylinder and hemisphere in part (a) are joined to form the solid in the diagram.
The solid is made of steel and 1 cm³ of steel has a mass of 7.85 g.
(i) Show that 1 cm³ of steel has a mass of 0.00785 kg.
(ii) Calculate the total mass of the solid.
(c) 2000 cm³ of iron is melted down and some of it is used to make 50 spheres with radius 2 cm.
(i) Calculate the percentage of iron that is left over.
[The volume, V, of a sphere with radius r is V = \(\frac{4}{3}\pi r^3\).]
(ii) The iron left over is then made into a cube.
Calculate the length of an edge of the cube.
(d) A solid cone has radius 3R cm and slant height 9R cm.
A solid cylinder has radius x cm and height 7x cm.
The total surface area of the cone is equal to the total surface area of the cylinder.
Given that R = kx, find the value of k.
[The curved surface area, A, of a cone with radius r and slant height l is A = πrl.]
▶️ Answer/Explanation
(a)(i) 1584 cm³
Volume of cylinder = πr²h = π × 6² × 14 = 1584 cm³ (using π = 3.142).
(a)(ii) 452 cm³
Volume of hemisphere = ½ × (4/3)πr³ = ½ × (4/3) × π × 6³ = 452 cm³.
(b)(i) 0.00785 kg
Convert grams to kilograms: 7.85 g = 7.85 ÷ 1000 = 0.00785 kg.
(b)(ii) 16.0 kg
Total volume = 1584 + 452 = 2036 cm³. Mass = 2036 × 0.00785 = 15.98 ≈ 16.0 kg.
(c)(i) 16.2%
Volume of 50 spheres = 50 × (4/3)π × 2³ = 1675.5 cm³. Leftover = 2000 – 1675.5 = 324.5 cm³. Percentage = (324.5/2000) × 100 = 16.2%.
(c)(ii) 6.87 cm
Cube root of leftover volume: ∛324.5 ≈ 6.87 cm.
(d) k = 2/3
Cone surface area = π(3R)² + π(3R)(9R) = 36πR². Cylinder surface area = 2πx² + 2πx(7x) = 16πx². Set equal: 36πR² = 16πx² → R/x = √(16/36) = 2/3.
(a) A cone has base diameter 8 cm and perpendicular height 15 cm.
(i) Calculate the volume of the cone.
(ii) A label completely covers the curved surface area of the cone. Calculate the area of the label as a percentage of the total surface area of the cone.
(b) An empty cylindrical container has radius 0.45 m. 300 litres of water is poured into the container at a rate of 375 ml per second.
(i) Find the time taken, in minutes and seconds, for all the water to be poured into the container.
(ii) Calculate the height of the water in the container.
▶️ Answer/Explanation
(a)(i) 251 or 251.3 to 251.4 cm³
Using V = (1/3)πr²h with r=4cm, h=15cm → V ≈ 251.3 cm³
(a)(ii) 79.5 or 79.51%
1. Find slant height (l) = √(4²+15²) ≈ 15.52cm
2. Curved area = π×4×15.52 ≈ 201.1cm²
3. Total area = 201.1 + π×4² ≈ 226.2cm²
4. Percentage = (201.1/226.2)×100 ≈ 79.5%
(b)(i) 13 min 20 sec
1. Convert 300L → 300,000ml
2. Time = 300,000 ÷ 375 = 800 seconds
3. Convert to minutes: 800÷60 = 13min 20sec
(b)(ii) 0.472 or 0.4715 to 0.4716 m
1. Convert 300L → 0.3m³
2. Using V = πr²h → 0.3 = π×(0.45)²×h
3. Solve for h ≈ 0.4715m