All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
2801
What is the volume of the largest sphere that can be carved out of a cube of edge 6 cm? (Leave answer in terms of π)
Answer:
36π cm³
The diameter of the largest sphere equals the edge of the cube. Diameter = 6 cm, so radius r = 3 cm. Volume = (4/3)πr³ = (4/3)π(3³) = (4/3)π(27) = 36π cm³.
2802
A right circular cylinder is circumscribing a hemisphere. What is the ratio of the volume of the cylinder to the volume of the hemisphere?
Answer:
3:2
If a cylinder circumscribes a hemisphere, its base radius is r and its height is equal to the radius of the hemisphere, h = r. Volume of cylinder = πr²h = πr³. Volume of hemisphere = (2/3)πr³. Ratio = πr³ / ((2/3)πr³) = 1 / (2/3) = 3/2, or 3:2.
2803
If the areas of the circular bases of a cylinder and a cone are equal, and their volumes are equal, what is the ratio of their heights (cylinder to cone)?
Answer:
1:3
Volume of cylinder = πr²h₁. Volume of cone = (1/3)πr²h₂. Since they are equal: πr²h₁ = (1/3)πr²h₂. Dividing by πr² gives h₁ = (1/3)h₂, so h₁ / h₂ = 1/3. The ratio is 1:3.
2804
If the surface area of a sphere is 4π cm², what is its volume?
Answer:
4/3 π cm³
Surface area = 4πr² = 4π, so r² = 1, meaning r = 1 cm. The volume is (4/3)πr³ = (4/3)π(1)³ = 4/3 π cm³.
2805
A spherical ball of lead 3 cm in radius is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm respectively. Find the radius of the third ball.
Answer:
2.5 cm
Volume is conserved. (4/3)π(3)³ = (4/3)π(1.5)³ + (4/3)π(2)³ + (4/3)π(r)³. Canceling (4/3)π gives: 27 = 3.375 + 8 + r³. 27 = 11.375 + r³. r³ = 15.625. Since 2.5³ = 15.625, the radius of the third ball is 2.5 cm.
2806
If the diameter of the base of a cone is 10 cm and its height is 12 cm, find its total surface area. (Use π = 3.14)
Answer:
282.6 cm²
Radius r = 5 cm, height h = 12 cm. Slant height l = √(5² + 12²) = 13 cm. Total surface area = πr(r + l) = 3.14 × 5 × (5 + 13) = 3.14 × 5 × 18 = 3.14 × 90 = 282.6 cm².
2807
A cone and a cylinder have equal bases and equal heights. The volume of the cone is V. What is the volume of the cylinder?
Answer:
3V
Volume of a cone = (1/3) × base area × height. Volume of a cylinder = base area × height. Therefore, the volume of the cylinder is exactly 3 times the volume of the cone. If cone volume is V, cylinder volume is 3V.
2808
A solid rectangular block of dimensions 4.4 m, 2.6 m, and 1 m is cast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
Answer:
112 m
Volume of block = 4.4 × 2.6 × 1 = 11.44 m³. Internal radius r = 30 cm = 0.3 m. External radius R = 30 + 5 = 35 cm = 0.35 m. Volume of hollow pipe = πh(R² - r²) = (22/7) × h × (0.35² - 0.30²) = (22/7) × h × (0.1225 - 0.09) = (22/7) × h × 0.0325. Equating volumes: (22/7) × h × 0.0325 = 11.44. h = (11.44 × 7) / (22 × 0.0325) = 80.08 / 0.715 = 112 m.
2809
What is the volume of the largest right circular cone that can be cut out of a cube of edge 14 cm? (Use π = 22/7)
Answer:
718.66 cm³
The base of the largest cone will touch the sides of the cube, so its diameter is 14 cm, giving a radius r = 7 cm. Its height will be equal to the edge of the cube, h = 14 cm. Volume = (1/3)πr²h = (1/3) × (22/7) × 7² × 14 = (1/3) × 22 × 7 × 14 = 2156 / 3 ≈ 718.66 cm³.
2810
A metallic sphere of radius 6 cm is melted and drawn into a wire of uniform circular cross-section. If the length of the wire is 72 m, find its radius.
Answer:
2 mm
Volume of sphere = (4/3)π(6)³ = 288π cm³. The wire is a cylinder. Let its radius be r cm. Length h = 36 m = 3600 cm. Volume of wire = πr²h = πr²(3600). Equating volumes: 3600πr² = 288π. r² = 288 / 3600 = 8 / 100 = 0.08. Wait, 288/3600 = 8/100 is wrong. 288/36 = 8, so 288/3600 = 8/100. Wait, √0.08 is not an integer. Let me re-calculate: V_sphere = 4/3 * π * 216 = 288π. V_wire = π * r² * 3600. r² = 288/3600 = 144/1800 = 72/900 = 36/450 = 4/50 = 2/25 = 0.08. That gives an irrational radius. Let me adjust the problem to radius 2 mm (0.2 cm). r² = 0.04. Then 3600 * 0.04 = 144. So sphere volume should be 144π, meaning R_sphere = ³√(108) which is not integer. Let me select an option close or change the length. If h = 36m, R=6cm. r² = 288/3600 = 0.08 cm² = 8 mm² -> r = √8 mm. Let me fix the height to 36 m? No, if V=288π. Let h = 72 m = 7200 cm. r² = 288/7200 = 4/100 = 0.04 cm². So r = 0.2 cm = 2 mm. I will assume the question intended length to be 72 m for option B. But since I can't edit the question easily, let me re-evaluate 288 / 3600 = 0.08. r = √0.08 cm = √8 mm ≈ 2.82 mm. I will rewrite the question to length = 72m.