Question 1

A cistern 6m long and 4 m wide contains water up to a depth of 1 m 25 cm. The total area of the wet surface is:

Accepted Answer

Problem Restatement Find the wetted surface area (bottom plus side walls up to the water level) of a rectangular cistern. Given data Length L = 6 m Width W = 4 m Water depth h = 1.25 m Concept / Approach Wet surface area = bottom area + area of four side walls in contact with water. Step-by-step calculation Bottom area = L × W = 6 × 4 = 24 m^2Side walls area = Perimeter × depth = 2(L + W) × h= 2(6 + 4) × 1.25 = 20 × 1.25 = 25 m^2Total wet surface area = 24 + 25 = 49 m^2 Common pitfalls Including the top surface (free surface) is incorrect. For sides, using Lh + Wh once; it must be twice each (four walls), or use perimeter. Final Answer Total wet surface area = 49 m^2.

Question 2

A boat having a length 3 m and breadth 2 m is floating on a lake. The boat sinks by 1 cm when a man gets on it. The mass of the man is:

Accepted Answer

Problem Restatement Determine the man's mass from the additional volume of water displaced when the boat sinks by 1 cm. Given data Boat plan dimensions: length = 3 m, breadth = 2 m (area = 6 m^2) Sinking (extra draft) = 1 cm = 0.01 m Density of water = 1000 kg/m^3 Concept / Approach By Archimedes' principle, additional weight supported = weight of extra water displaced. Mass = density × displaced volume. Step-by-step calculation Extra displaced volume = Area × Extra draft = 6 × 0.01 = 0.06 m^3Mass of displaced water = 1000 × 0.06 = 60 kg Final Answer Mass of the man = 60 kg.

Question 3

How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to build a wall of 8 m x 6 m x 22.5 cm?

Accepted Answer

Problem Restatement Compute the number of identical bricks required to fill the volume of a wall. Given data Brick size = 25 cm × 11.25 cm × 6 cm Wall size = 8 m × 6 m × 22.5 cm Concept / Approach Number of bricks = (Volume of wall) ÷ (Volume of one brick). Use consistent units (convert metres to centimetres). Step-by-step calculation Convert wall dimensions: 8 m = 800 cm; 6 m = 600 cm; thickness = 22.5 cmWall volume = 800 × 600 × 22.5 = 10,800,000 cm^3Brick volume = 25 × 11.25 × 6 = 1687.5 cm^3Number of bricks = 10,800,000 ÷ 1687.5 = 6400 Notes Mortar and wastage are ignored as instructed. Final Answer Required bricks = 6400.

Question 4

A right triangle with sides 3 cm, 4 cm and 5 cm is rotated the side of 3 cm to form a cone. The volume of the cone so formed is:

Accepted Answer

Problem Restatement Rotating a right triangle about one leg forms a right circular cone: the axis (height) is the leg of rotation and the radius is the other leg. Given data Right triangle sides: 3 cm, 4 cm, 5 cm (hypotenuse = 5 cm) Rotation about the side of 3 cm Concept / Approach Here, height h = 3 cm (axis of rotation) and radius r = 4 cm (the other perpendicular leg). The slant height (5 cm) is not needed for volume. Step-by-step calculation Cone volume V = (1/3)πr^2h= (1/3)π(4^2)(3) = (1/3)π × 16 × 3 = 16π cm^3 Final Answer Volume of the cone = 16π cm^3.

Question 5

The curved surface area of a cylindrical pillar is 264 m2 and its volume is 924 m3. Find the ratio of its diameter to its height.

Accepted Answer

Problem Restatement For a cylinder, use CSA = 2πrh and Volume = πr^2h to determine the ratio diameter : height. Given formulas Curved surface area: 2πr h = 264 Volume: πr^2 h = 924 Concept / Approach Eliminate h or r using the two equations to find dimensions up to a constant and then form the ratio. Step-by-step calculation From volume: πr^2h = 924 → h = 924 ÷ (πr^2)Substitute in CSA: 2πr × [924 ÷ (πr^2)] = 264Simplify: 2 × 924 ÷ r = 264 → 1848 ÷ r = 264 → r = 1848 ÷ 264 = 7Radius r = 7 m → Diameter d = 14 mFind height: πr^2h = 924 → π(49)h = 924 → h = 924 ÷ (49π) ≈ 6Therefore, diameter : height = 14 : 6 = 7 : 3 Check CSA = 2πr h = 2&pi(7)(6) = 84π ≈ 264 ✓ Final Answer Ratio (diameter : height) = 7 : 3.

Question 6

What is the total surface area of a right circular cone whose height is 14 cm and base radius is 7 cm?

Accepted Answer

Problem Restatement Find the total surface area (TSA) of a right circular cone given its height and base radius. Given data Height, h = 14 cm Radius, r = 7 cm Slant height, l = √(r^2 + h^2) Concept / Formula Slant height: l = √(r^2 + h^2) Total surface area: TSA = πr(l + r) Step-by-step calculation l = √(7^2 + 14^2) = √(49 + 196) = √245 = 7√5 ≈ 15.652TSA = π × 7 × (15.652 + 7)Using π = 22/7 → TSA = (22/7) × 7 × 22.652 = 22 × 22.652 = 498.35 cm^2 (approx.) Checks & Notes Using exact form: TSA = 49π(√5 + 1) cm^2. Always include both lateral area and base area when TSA is asked. Final Answer Total surface area ≈ 498.35 cm^2.

Question 7

On a religious day, 50 men take a dip in a rectangular water tank that is 40 m long and 20 m wide. If each man displaces, on average, 4 m^3 of water, by how much does the water level in the tank rise?

Accepted Answer

Problem Restatement Compute the rise in water level when a known total volume is added to a tank of known plan area. Given data Number of men = 50 Average displacement per man = 4 m^3 Tank length = 40 m, breadth = 20 m → plan area A = 40 × 20 = 800 m^2 Concept / Formula Rise in level = Added volume ÷ Plan area. Step-by-step calculation Total displaced volume = 50 × 4 = 200 m^3Rise, h = 200 ÷ 800 = 0.25 m = 25 cm Final Answer Water level rises by 25 cm.

Question 8

Silver of volume 66 cm^3 is drawn into a cylindrical wire of diameter 1 mm. What is the length of the wire (in metres)?

Accepted Answer

Problem Restatement Treat the wire as a cylinder with known volume and cross-sectional diameter; find its length. Given data Volume V = 66 cm^3 Diameter d = 1 mm = 0.1 cm → radius r = 0.05 cm Concept / Formula For a cylinder: V = πr^2L Hence L = V ÷ (πr^2) Step-by-step calculation Area = πr^2 = π(0.05)^2 = π × 0.0025 ≈ 0.00785398 cm^2L = 66 ÷ 0.00785398 ≈ 8405.28 cm = 84.05 m (approx.) Rounding Length ≈ 84 m.

Question 9

A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling equals the sum of the areas of the four walls, what is the volume of the hall?

Accepted Answer

Problem Restatement Use the relation between floor/ceiling areas and wall areas to determine the height, then compute volume. Given data Length L = 15 m Breadth B = 12 m Unknown height = H Concept / Relations Floor area = Ceiling area = L × B Sum of floor and ceiling = 2LB Area of four walls = 2H(L + B) Step-by-step calculation Given: 2LB = 2H(L + B)2(15)(12) = 2H(27) → 360 = 54H → H = 360 ÷ 54 = 20/3 mVolume = L × B × H = 15 × 12 × (20/3) = 1200 m^3 Final Answer Volume of the hall = 1200 m^3.

Question 10

A cistern of capacity 8000 litres has external dimensions 3.3 m × 2.6 m × 1.1 m. Its side walls are 5 cm thick. What is the thickness of the bottom (assume the top is open)?

Accepted Answer

Problem Restatement Determine the bottom thickness using capacity (internal volume) and external dimensions with known wall thickness. Given data Capacity = 8000 L = 8 m^3 External L × B × H = 3.3 m × 2.6 m × 1.1 m Side wall thickness = 5 cm = 0.05 m Bottom thickness = x m (unknown) Concept / Setup Internal dimensions: Li = 3.3 − 2(0.05) = 3.2 m; Bi = 2.6 − 2(0.05) = 2.5 m; Hi = 1.1 − x (top open) Step-by-step calculation Internal volume = LiBiHi = 3.2 × 2.5 × (1.1 − x)8 = 8.0 × (1.1 − x) → 1.1 − x = 1.0 → x = 0.1 m = 10 cm Final Answer Bottom thickness = 10 cm.

Question 11

A large cube is formed by melting three smaller solid cubes of edges 3 cm, 4 cm, and 5 cm. What is the ratio of the total surface area of the three small cubes to that of the large cube?

Accepted Answer

Problem Restatement Use volume conservation to get the large cube's edge, then compare total surface areas. Given data Small cube edges: 3 cm, 4 cm, 5 cm Step 1: Find large cube edge Total volume = 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216 cm^3Large cube edge a satisfies a^3 = 216 → a = 6 cm Step 2: Compare total surface areas TSA (three small) = 6(3^2 + 4^2 + 5^2) = 6(9 + 16 + 25) = 6 × 50 = 300 cm^2TSA (large) = 6a^2 = 6 × 36 = 216 cm^2Required ratio = 300 : 216 = 25 : 18 Final Answer Ratio of areas = 25 : 18.

Question 12

A hollow iron pipe is 21 cm long with external diameter 8 cm and thickness 1 cm. If the density of iron is 8 g/cm^3, what is the weight of the pipe?

Accepted Answer

Problem Restatement Compute mass of a hollow cylindrical shell from its geometric volume and material density. Given data Length, L = 21 cm External diameter = 8 cm → R = 4 cm Thickness = 1 cm → inner radius r = 3 cm Density ρ = 8 g/cm^3 Concept / Formula Volume of material (hollow cylinder): V = πL(R^2 − r^2) Mass = ρV Step-by-step calculation V = π × 21 × (4^2 − 3^2) = π × 21 × 7 = 147π cm^3Using π = 22/7 → V = 147 × 22/7 = 462 cm^3Mass = 8 × 462 = 3696 g = 3.696 kg Final Answer Weight of the pipe ≈ 3.696 kg.

Question 13

During a shower, 5 cm of rain falls uniformly. What volume of water falls on 1.5 hectares of ground?

Accepted Answer

Problem Restatement Convert rainfall depth over an area into a volume. Given data Rainfall depth = 5 cm = 0.05 m Area = 1.5 hectares = 1.5 × 10,000 m^2 = 15,000 m^2 Concept / Formula Volume = Depth × Area Step-by-step calculation V = 0.05 × 15,000 = 750 m^3 Alternative unit = 750,000 litres (since 1 m^3 = 1000 L) Final Answer Volume of rainfall = 750 m^3.

Question 14

A rectangular metal sheet measures 48 m by 36 m. From each corner, a square of side 8 m is cut out and the sides are folded up to form an open box. What is the volume of the box (in m^3)?

Accepted Answer

Problem Restatement Find the volume of an open box formed by cutting equal squares from each corner of a rectangle and folding up the flaps. Given data Sheet: 48 m × 36 m Cut-out square side = 8 m Concept / Dimensions after folding New length = 48 − 2×8 = 32 m; New breadth = 36 − 2×8 = 20 m; Height = 8 m. Step-by-step calculation Volume = L × B × H = 32 × 20 × 8 = 5120 m^3 Pitfall Do not forget to reduce both length and breadth by twice the cut size. Final Answer Volume of the box = 5120 m^3.

Question 15

A right circular cone has slant height 10 m and vertical height 8 m. What is the area of its curved (lateral) surface?

Accepted Answer

Problem Restatement Compute the lateral surface area (CSA) of a cone given its slant height and vertical height. Given data Slant height, l = 10 m Vertical height, h = 8 m Concept / Formulae Radius from Pythagoras: r = √(l^2 − h^2) Curved surface area: CSA = πr l Step-by-step calculation r = √(10^2 − 8^2) = √(100 − 64) = √36 = 6 mCSA = π × 6 × 10 = 60π m^2 ≈ 188.50 m^2 Final Answer Curved surface area = 60π m^2 (≈ 188.5 m^2).

Question 16

A cube has volume 512 cm^3. Find its total surface area.

Accepted Answer

Introduction / Context:For a cube, volume and surface area are functions of the side length a. If the volume is known, find a as the cube root, then compute surface area as 6a^2. This is a straightforward application of cube geometry formulas. Given Data / Assumptions: Volume V = a^3 = 512 cm^3. Total surface area S = 6a^2. Concept / Approach:Take the cube root of 512 to find a; then square a and multiply by 6 to obtain S. Step-by-Step Solution:a = ∛512 = 8 cmS = 6a^2 = 6 * 8^2 = 6 * 64 = 384 cm^2 Verification / Alternative check:Compute volume from a = 8: 8^3 = 512, confirming side length. Why Other Options Are Wrong:64 cm^2 is one face area (a^2) times 1; 256 cm^2 is 4a^2; 512 cm^2 equals 8a^2; only 384 cm^2 equals 6a^2. Common Pitfalls:Accidentally using 6a instead of 6a^2; mixing units (keeping “cm” vs “cm^2”). Final Answer:384 cm2

Question 17

A cube has total surface area 726 m^2. Find its volume in m^3.

Accepted Answer

Introduction / Context:Given a cube’s total surface area S, find side a from S = 6a^2, then compute the volume V = a^3. This is the reverse of the previous problem type. Given Data / Assumptions: S = 726 m^2. S = 6a^2 ⇒ a^2 = S/6. V = a^3. Concept / Approach:Compute a^2, then a, then a^3. The arithmetic is neat because 726/6 is a perfect square. Step-by-Step Solution:a^2 = 726 / 6 = 121a = √121 = 11 mV = a^3 = 11^3 = 1331 m^3 Verification / Alternative check:Back-calc surface area: 6a^2 = 6 * 121 = 726 m^2, confirming a = 11 m. Why Other Options Are Wrong:Other numbers do not equal 11^3 and would correspond to non-integer sides or incorrect squares. Common Pitfalls:Taking a as 726/6 (forgetting the square root), or miscomputing 11^3. Final Answer:1331 m3

Question 18

If each side of a cube is doubled, how does its volume change?

Accepted Answer

Introduction / Context:Volume scales with the cube of the linear dimension. If all edges of a 3D object are scaled by a factor k, the volume scales by k^3. Here, each side is doubled, so k = 2, and the new volume is 2^3 = 8 times the original. Given Data / Assumptions: Original side length a. New side length = 2a. Volume formula for cube: V = a^3. Concept / Approach:Compare V′ = (2a)^3 to V = a^3; the ratio V′/V reveals the multiplicative change. Step-by-Step Calculation:V′ = (2a)^3 = 8a^3V′/V = 8a^3 / a^3 = 8 Verification / Alternative check:Try a = 1 ⇒ V = 1; doubling gives a = 2 ⇒ V′ = 8; ratio is 8, confirming. Why Other Options Are Wrong:2x, 4x, and 6x correspond to squaring or linear misconceptions; volume scales with the cube of linear change. Common Pitfalls:Confusing area scaling (k^2) with volume scaling (k^3). Final Answer:Becomes 8 times

Question 19

Three metal cubes with sides 5 cm, 4 cm, and 3 cm are melted and recast into a single cube. Find the edge length of the new cube.

Accepted Answer

Introduction / Context:Melt-and-recast problems conserve volume (neglecting loss). The sum of the original volumes equals the volume of the new cube. Therefore, find the total initial volume and take its cube root to get the new edge. Given Data / Assumptions: Original cube edges: 5 cm, 4 cm, 3 cm. Volumes: 125, 64, and 27 cm^3 respectively. Conservation of volume: V_new = 125 + 64 + 27. Concept / Approach:Compute V_total = 5^3 + 4^3 + 3^3 = 216 cm^3, then solve a^3 = 216 for a. Step-by-Step Solution:V_total = 125 + 64 + 27 = 216 cm^3a = ∛216 = 6 cm Verification / Alternative check:6^3 = 216; volumes balance exactly. Why Other Options Are Wrong:8 cm would require 512 cm^3; 10 cm requires 1000 cm^3; both exceed the available metal volume. Common Pitfalls:Adding side lengths instead of volumes; forgetting that volume scales as the cube of edge length. Final Answer:6 cm

Question 20

Two cubes have volumes in the ratio 8 : 1. Find the ratio of their edges.

Accepted Answer

Introduction / Context:For similar solids (including cubes), volume scales as the cube of the linear dimension. If V1 : V2 = 8 : 1, then (a1/a2)^3 = 8/1, so the edge-length ratio is the cube root of the volume ratio. Given Data / Assumptions: V1 : V2 = 8 : 1. V ∝ a^3 for cubes. Concept / Approach:Take cube roots of both sides of the ratio: (a1/a2) = ∛(8/1) = 2/1. Step-by-Step Solution:(a1/a2)^3 = 8 ⇒ a1/a2 = 2 Verification / Alternative check:Let a2 = 1 ⇒ V2 = 1; set a1 = 2 ⇒ V1 = 8; ratio holds. Why Other Options Are Wrong:8 : 1 is a volume ratio; 2√2 : 1 is irrelevant; “None” is unnecessary because 2 : 1 is correct. Common Pitfalls:Squaring or leaving the ratio unchanged instead of taking cube roots. Final Answer:2 : 1

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