Question 1

Right circular cylinder – total surface area (TSA): A cylinder has base diameter 14 cm (radius 7 cm) and height 40 cm. Find its total surface area (both circular ends + curved surface) in cm^2.

Accepted Answer

Introduction / Context: Total surface area (TSA) of a cylinder sums the curved area and the areas of the two circular bases. This tests recall of the TSA formula and careful substitution of radius and height. Given Data / Assumptions: Diameter = 14 cm → r = 7 cm Height h = 40 cm TSA = 2πr(h + r) Concept / Approach: Use the compact TSA identity to avoid computing parts separately. Keep π symbolic until the final step to minimize rounding error, then evaluate numerically to compare with the options. Step-by-Step Solution: TSA = 2π * 7 * (40 + 7) = 14π * 47 = 658π cm^2Using π ≈ 3.14 → TSA ≈ 658 * 3.14 ≈ 2066.12 cm^2Using π = 22/7 → TSA = 658 * (22/7) = 94 * 22 = 2068 cm^2 Verification / Alternative check: Compute parts: curved = 2πrh = 2π * 7 * 40 = 560π; two bases = 2πr^2 = 2π * 49 = 98π; sum = 658π → consistent. Why Other Options Are Wrong: 1825, 1925, 2050, 2160 do not equal 658π; 2068 matches exactly with π = 22/7, a standard exam convention. Common Pitfalls: Using diameter instead of radius in r^2; omitting one base; mixing curved surface area with TSA. Final Answer: 2068 sq cm

Question 2

Cylinder – find radius from curved (lateral) surface area: A cylinder has lateral surface area (CSA) 94.2 cm^2 and height 5 cm. Using π = 3.14, compute the radius of its base.

Accepted Answer

Introduction / Context: The curved (lateral) surface area of a cylinder is the rectangle that wraps around: width equals the height h and length equals the circumference 2πr. This question checks algebraic manipulation of the CSA formula to recover r from given CSA and h. Given Data / Assumptions: CSA = 94.2 cm^2 Height h = 5 cm π = 3.14 CSA formula: 2πrh Concept / Approach: Rearrange 2πrh = CSA to r = CSA / (2πh). Substitute numeric values carefully and simplify. Step-by-Step Solution: 2πh = 2 * 3.14 * 5 = 31.4r = 94.2 / 31.4 = 3.0 cm Verification / Alternative check: Plug back: 2πrh = 2 * 3.14 * 3 * 5 = 94.2 cm^2, matching exactly. Why Other Options Are Wrong: 5 cm and 4 cm make CSA too large; 8 cm is far too large; 2 cm would give only 62.8 cm^2, not 94.2 cm^2. Common Pitfalls: Using πr^2h (volume) instead of 2πrh (CSA); multiplying by diameter in place of radius; rounding too early. Final Answer: 3 cm

Question 3

Volume conservation – rod hammered into wire: A solid circular rod of diameter 2 cm and length 30 cm is drawn into a uniform wire of length 3 m (i.e., 300 cm). Assuming no loss of material, find the diameter of the wire (in cm).

Accepted Answer

Introduction / Context: When a metal rod is drawn into a wire, the volume of the material remains constant (neglecting losses). For cylinders, volume equals πr^2h. Equating the initial and final cylinder volumes allows recovery of the unknown final radius/diameter from the change in length. Given Data / Assumptions: Initial: diameter 2 cm → r1 = 1 cm; length L1 = 30 cm Final: length L2 = 3 m = 300 cm; radius r2 unknown; diameter = 2r2 Volume conservation: πr1^2L1 = πr2^2L2 Concept / Approach: Cancel π and solve for r2^2 = (r1^2 * L1) / L2. Then take square root to get r2 and multiply by 2 for the diameter. Step-by-Step Solution: r2^2 = (1^2 * 30) / 300 = 0.1r2 = √0.1 = 1/√10 cmDiameter = 2r2 = 2/√10 cm Verification / Alternative check: Check volumes: initial V1 = π * 1^2 * 30 = 30π; final V2 = π * (1/√10)^2 * 300 * (2^2?) (careful: radius used above) → using r2^2 = 0.1 gives V2 = π * 0.1 * 300 = 30π, matching V1. Why Other Options Are Wrong: 2/10 cm = 0.2 cm and 1/10 cm = 0.1 cm do not satisfy volume equality; 1/√10 cm is the radius, not the diameter; 1/5 cm is inconsistent with the computed r2. Common Pitfalls: Mixing up diameter and radius; forgetting to convert 3 m to 300 cm; incorrectly applying √ to the ratio; failing to conserve volume. Final Answer: 2/√10 cm

Question 4

Find cylinder height from total surface area: A solid cylinder has radius 5 cm and total surface area 660 cm^2. Compute its height h (in cm).

Accepted Answer

Introduction / Context: Total surface area (TSA) for a cylinder equals the curved surface area plus the areas of both circular ends. Given TSA and radius, we can isolate height by algebra. This tests formula fluency and neat handling of π in denominators. Given Data / Assumptions: r = 5 cm TSA = 660 cm^2 TSA formula: 2πr(h + r) Concept / Approach: Solve 2πr(h + r) = 660 for h. Keep the algebra symbolic until the last step to avoid rounding drift; then evaluate numerically with π ≈ 3.14 (or 22/7) and round to a listed option. Step-by-Step Solution: 2πr(h + r) = 660 → 10π(h + 5) = 660h + 5 = 660 / (10π) = 66 / πUsing π ≈ 3.14 → h + 5 ≈ 21.019 → h ≈ 16.019 cm ≈ 16 cm Verification / Alternative check: Plug h = 16 into TSA: 2π * 5 * (16 + 5) = 10π * 21 = 210π ≈ 659.73 cm^2, which rounds to 660 cm^2. Why Other Options Are Wrong: 10, 12, 14, 15 cm give TSA values significantly apart from 660 cm^2 when substituted into the formula; 16 cm uniquely satisfies the equation within rounding. Common Pitfalls: Using curved area 2πrh in place of TSA; forgetting both ends; arithmetic slips when dividing by π. Final Answer: 16 cm

Question 5

Cylinder – curved surface area from radius and height: Find the curved (lateral) surface area of a right circular cylinder with radius 7 cm and height 160 cm. Give the answer in cm^2.

Accepted Answer

Introduction / Context: The curved surface area (CSA) of a cylinder equals the area of a rectangle with height h and width equal to the circumference 2πr. This is a direct substitution exercise once r and h are identified correctly. Given Data / Assumptions: r = 7 cm h = 160 cm CSA = 2πrh Concept / Approach: Compute 2 * π * 7 * 160 and evaluate numerically, preferably with π = 22/7 to get an exact integer in this setup (common in aptitude problems). Step-by-Step Solution: CSA = 2πrh = 2 * π * 7 * 160 = 2240π cm^2With π = 22/7 → CSA = 2240 * (22/7) = 320 * 22 = 7040 cm^2 Verification / Alternative check: Using π ≈ 3.14 yields 2240 * 3.14 ≈ 7033.6 cm^2, which rounds to 7034–7040; consistent with the exact 22/7 value. Why Other Options Are Wrong: 6020 and 5052 are too small; 7045 and 7080 overshoot the precise value; 7040 cm^2 matches the exact arithmetic with π = 22/7. Common Pitfalls: Using diameter instead of radius; confusing TSA with CSA; arithmetic errors multiplying large factors. Final Answer: 7040 sq cm

Question 6

Ratio of volumes of two cylinders: Two cylinders have radii in the ratio 2 : 3 and heights in the ratio 5 : 3. Find the ratio of their volumes (first : second).

Accepted Answer

Introduction / Context: The volume of a cylinder scales as r^2h. For similar or related solids, dimensional ratios translate into volume ratios by squaring the radius ratio and multiplying by the height ratio. This checks proportional reasoning. Given Data / Assumptions: r1 : r2 = 2 : 3 h1 : h2 = 5 : 3 V ∝ r^2h Concept / Approach: Compute (r1^2 : r2^2) = (2^2 : 3^2) = (4 : 9) and multiply by (h1 : h2) = (5 : 3). The resulting ratio is (4*5 : 9*3) = (20 : 27). Step-by-Step Solution: r^2 ratio = 4 : 9Multiply by h ratio → 4*5 : 9*3 = 20 : 27 Verification / Alternative check: Pick convenient numbers: let r1 = 2, r2 = 3, h1 = 5, h2 = 3. Then V1 = π * 2^2 * 5 = 20π; V2 = π * 3^2 * 3 = 27π → ratio 20 : 27. Why Other Options Are Wrong: 4:9 and 9:4 ignore height; 27:20 reverses order; 10:9 is unrelated to r^2h scaling. Common Pitfalls: Forgetting to square the radius ratio; mixing up which cylinder is “first” in the final ratio. Final Answer: 20 : 27

Question 7

Cylindrical pillar – find diameter : height ratio from CSA and volume: A cylindrical pillar has curved surface area 264 m^2 and volume 924 m^3. Find the ratio of its diameter to its height.

Accepted Answer

Introduction / Context: Given CSA and volume of a cylinder, we can express height h in terms of r using CSA, then substitute into the volume formula to solve for r. Once r is known, diameter : height follows directly. This combines two standard cylinder relations. Given Data / Assumptions: CSA = 2πrh = 264 Volume V = πr^2h = 924 Unknowns: r (m) and h (m) Concept / Approach: From CSA: h = 264 / (2πr) = 132 / (πr). Substitute into V to eliminate h and solve r. Then compute h explicitly and take (diameter):(height) = (2r):h. Step-by-Step Solution: h = 132 / (πr)V = πr^2h = πr^2 * (132 / (πr)) = 132r = 924 → r = 924 / 132 = 7 mh = 132 / (π * 7) ≈ 132 / 21.991 ≈ 6 mDiameter = 2r = 14 mRatio (diameter : height) ≈ 14 : 6 = 7 : 3 Verification / Alternative check: Plug r = 7, h ≈ 6: CSA = 2π * 7 * 6 = 84π ≈ 263.9 ≈ 264; Volume = π * 49 * 6 ≈ 924, both consistent. Why Other Options Are Wrong: 3:7 and 6:7 invert or mismatch the derived ratio; 7:6 assumes h = 12 rather than ≈ 6; 14:5 overstates the diameter relative to height. Common Pitfalls: Forgetting that CSA uses 2πrh; solving volume first without eliminating h; rounding too early. Final Answer: 7 : 3

Question 8

Effect on cylinder volume when height changes: If a cylinder’s height decreases by 8% while its radius is unchanged, by what percentage does its volume change?

Accepted Answer

Introduction / Context: Volume of a cylinder is V = πr^2h. When only one dimension changes, the percentage change in V equals the percentage change in that dimension’s factor (holding r constant). This is an application of proportional reasoning. Given Data / Assumptions: r unchanged h decreases by 8% → new h = 0.92h V_new = πr^2 * (0.92h) = 0.92 * V_old Concept / Approach: Because V is directly proportional to h (with r fixed), the volume decreases by the same 8%. No computation with π or r is needed beyond recognizing proportionality. Step-by-Step Solution: V_old = πr^2hV_new = πr^2 * (0.92h) = 0.92 V_oldPercentage decrease = (1 − 0.92) * 100% = 8% Verification / Alternative check: Try numbers: let r = 1, h = 100. V_old = 100π. New h = 92 → V_new = 92π. Decrease = 8π out of 100π = 8%. Why Other Options Are Wrong: 4% and 6% understate; 10% overstates; 12% is unrelated. Only 8% matches the direct dependence on h. Common Pitfalls: Thinking area or volume changes require squaring or cubing in every case; here only height changes, so the effect is linear and equal to 8%. Final Answer: 8%

Question 9

Effect on cylinder volume when radius increases: If the cylinder’s radius increases by 25% while height stays the same, what is the percentage increase in volume?

Accepted Answer

Introduction / Context: Volume depends on r^2h. With height fixed, the percent change in V is governed by the square of the radius scale factor. This tests understanding of quadratic dependence on r. Given Data / Assumptions: r increases by 25% → r_new = 1.25r h unchanged V_new / V_old = (r_new^2 / r^2) = (1.25)^2 = 1.5625 Concept / Approach: Compute the area factor from the squared radius ratio, then convert the factor to a percentage increase: (1.5625 − 1) × 100%. Step-by-Step Solution: Scale factor in r = 1.25Scale factor in r^2 = 1.25^2 = 1.5625Percent increase = 0.5625 × 100% = 56.25% Verification / Alternative check: Pick r = 4, h = 10: V_old = π * 16 * 10 = 160π. New r = 5 → V_new = π * 25 * 10 = 250π. Increase = 90π → 90/160 = 56.25% increase. Why Other Options Are Wrong: 52.25% and 50.4% are not equal to (1.25^2 − 1); 60.26% is an overestimate; 25% confuses linear with quadratic dependence. Common Pitfalls: Applying 25% directly to volume; forgetting the square on r in πr^2h. Final Answer: 56.25%

Question 10

Combined changes in radius and height – effect on volume: A cylinder’s radius decreases by 8% while its height increases by 4%. What is the net percentage change in volume?

Accepted Answer

Introduction / Context: Volume V = πr^2h. Independent percentage changes in r and h multiply as factors: r contributes quadratically (r^2), h linearly. This question checks compound percentage reasoning with multiplication of factors. Given Data / Assumptions: r_new = 0.92r (8% decrease) h_new = 1.04h (4% increase) V_new / V_old = (0.92^2) * (1.04) Concept / Approach: Compute the squared radius factor first, then multiply by the height factor, and finally convert to a percent change relative to 1 (100%). Step-by-Step Solution: 0.92^2 = 0.8464Overall factor = 0.8464 * 1.04 = 0.880256Percent change = (0.880256 − 1) × 100% = −11.9744% Verification / Alternative check: Numerical sanity: a ~12% decrease is expected since the reduction in r^2 (≈ −15.36%) dominates the modest +4% height increase, yielding around −11.96%. Why Other Options Are Wrong: Increase options contradict the computed factor < 1; 12.4678% refers to a different hypothetical combination; 8% is relevant only if h stayed constant. Common Pitfalls: Adding percentages linearly (−16% + 4% = −12%) without squaring r; forgetting multiplicative compounding. Final Answer: 11.9744% (decrease)

Question 11

Painting four walls – find hall height from cost and perimeter: A rectangular hall has floor perimeter 250 m. Painting the four walls costs ₹ 15000 at ₹ 10 per m^2. Neglect openings and find the height of the hall.

Accepted Answer

Introduction / Context: Painting the four walls covers the lateral surface area of the room: area = perimeter of floor × height. Given cost and rate, we can recover area, then divide by perimeter to find height. This is a practical application of unit rates and surface area of prisms. Given Data / Assumptions: Perimeter P = 250 m Cost = ₹ 15000 Rate = ₹ 10 per m^2 → painted area A = 15000 / 10 = 1500 m^2 A = P × h Concept / Approach: Use the formula for the lateral area of a rectangular prism (room without ceiling/floor): it equals the floor perimeter multiplied by wall height. Solve h = A / P. Step-by-Step Solution: A = 15000 / 10 = 1500 m^2h = A / P = 1500 / 250 = 6 m Verification / Alternative check: If L + W = 125 m (since 2(L + W) = 250), then total wall area = 2h(L + W) * 2? (equivalently P * h) = 250h → with h = 6, area = 1500 m^2; cost = 1500 * 10 = ₹ 15000, consistent. Why Other Options Are Wrong: Heights 5, 7, 8, 9 m would give painted areas 1250, 1750, 2000, 2250 m^2 respectively, not matching the billed area of 1500 m^2. Common Pitfalls: Attempting to use room length/width individually (they are unnecessary); forgetting that ceilings/floors are excluded in “four walls only.” Final Answer: 6 m

Question 12

Ground rolled by a cylindrical roller – area covered: A roller has diameter 84 cm and length 120 cm. It makes 500 complete revolutions to level a playground. Find the total area rolled (in m^2).

Accepted Answer

Introduction / Context: Each revolution of a cylindrical roller covers a rectangular strip whose width equals the roller length and whose length equals the circumference of the circular cross-section. The total area equals (area per revolution) × (number of revolutions). Given Data / Assumptions: Diameter d = 84 cm = 0.84 m → circumference C = πd Length L = 120 cm = 1.2 m Revolutions n = 500 Area per revolution = C * L Concept / Approach: Compute area = n * (πd * L). Using π = 22/7 simplifies the arithmetic here to an exact multiple, a common exam design. Step-by-Step Solution: Per revolution area = π * 0.84 * 1.2 = π * 1.008 m^2Using π = 22/7 → per rev = (22/7) * 1.008 = (22 * 1.008) / 7 ≈ 3.168 m^2Total area = 500 * 3.168 = 1584 m^2 Verification / Alternative check: Work in centimeters: C = π * 84; strip width = 120; per rev area = 84π * 120 cm^2 = 10080π cm^2; times 500 → 5,040,000π cm^2. Divide by 10,000 to convert to m^2 → 504π ≈ 1584 (since π ≈ 22/7). Why Other Options Are Wrong: 1632 and 1532 are nearby but not equal to 504π; 1817 is too large; 1600 is a rounded guess, not the exact value. Common Pitfalls: Using radius instead of diameter in circumference; forgetting unit conversion cm ↔ m; overlooking that area per revolution is linear in both circumference and roller length. Final Answer: 1584

Question 13

Cost to paint the curved surface of a cylindrical pillar: A pillar of diameter 50 cm (radius 0.25 m) and height 3.5 m is painted on its curved surface at ₹ 10 per m^2. Find the cost.

Accepted Answer

Introduction / Context: Painting the curved surface of a cylinder uses the lateral area 2πrh. Multiply the area by the rate to get the total cost. This exercise checks unit conversion and correct identification of “covered surface” as the curved area only. Given Data / Assumptions: Diameter = 0.50 m → r = 0.25 m Height h = 3.5 m Rate = ₹ 10 per m^2 Curved area A = 2πrh Concept / Approach: Compute A first, then multiply by rate. Use π ≈ 3.14 (or 22/7) for a numeric amount in rupees; round to the nearest rupee if needed. Step-by-Step Solution: A = 2πrh = 2 * π * 0.25 * 3.5 = 1.75π m^2Using π ≈ 3.14 → A ≈ 5.495 m^2Cost = 5.495 * ₹ 10 ≈ ₹ 54.95 ≈ ₹ 55 Verification / Alternative check: Using π = 22/7 gives A = 1.75 * (22/7) = 5.5 m^2 → cost ₹ 55 exactly. Both calculations agree within rounding. Why Other Options Are Wrong: ₹ 50 and ₹ 60 bracket the correct value but are off; ₹ 68 and ₹ 98 are too large given the modest area; ₹ 55 uniquely matches the computation. Common Pitfalls: Including top/bottom surfaces (not asked); leaving diameter instead of radius in 2πrh; forgetting to convert 50 cm to meters when needed. Final Answer: ₹ 55

Question 14

Volume of cylindrical pillar (material used): A solid circular pillar has diameter 14 cm (radius 7 cm) and height 5 m (i.e., 500 cm). Assuming it is a solid cylinder, find its volume in cm^3.

Accepted Answer

Introduction / Context: The volume of a solid cylindrical pillar is V = πr^2h. Dimensions are given with mixed units (cm and m), so convert consistently before computation. Present the exact result using π = 22/7 to get a neat factorization in the provided format. Given Data / Assumptions: Diameter = 14 cm → r = 7 cm Height h = 5 m = 500 cm V = πr^2h Concept / Approach: Keep units in centimeters to match the requested cm^3 output. Using π = 22/7 typically simplifies multiples involving 7 in r. Step-by-Step Solution: V = π * 7^2 * 500 = π * 49 * 500 = 24500π cm^3With π = 22/7 → V = 24500 * (22/7) = 3500 * 22 = 77000 cm^3Expressed as 77 × 10^3 cm^3 Verification / Alternative check: Rough check with π ≈ 3.14: 24500π ≈ 76930 cm^3, consistent with 77000 when π = 22/7 is used by design. Why Other Options Are Wrong: (77 × 10^2) and (77 × 10^4) are off by factors of 10; (77 × 10^5) and (77 × 10^6) are orders of magnitude too large. Common Pitfalls: Not converting 5 m to 500 cm; using diameter in r^2; rounding mid-calculation instead of at the end. Final Answer: (77 x 10^3)cm3

Question 15

Right circular cone – find slant height from curved surface area: A cone has radius 14 cm and curved surface area 440 cm^2. Compute its slant height l (in cm).

Accepted Answer

Introduction / Context: The curved (lateral) surface area of a right circular cone equals πrl, where r is the base radius and l is the slant height. This item checks rearrangement of the CSA formula to isolate l. Given Data / Assumptions: CSA = 440 cm^2 r = 14 cm Use π = 22/7 (yields an exact result here) CSA formula: πrl Concept / Approach: From πrl = 440, solve l = 440 / (πr). With r a multiple of 7, π = 22/7 simplifies nicely to an integer value for l. Step-by-Step Solution: Denominator πr = (22/7) * 14 = 44l = 440 / 44 = 10 cm Verification / Alternative check: Plugging back: πrl = (22/7) * 14 * 10 = 440 cm^2, exact match. Why Other Options Are Wrong: 11, 12, 13, 14 cm do not satisfy πrl = 440 with r = 14; only 10 cm does. Common Pitfalls: Using πr^2 (area of circle) instead of πrl; confusing height with slant height. Final Answer: 10 cm

Question 16

Right circular cone – lateral surface area: A cone has base diameter 7 cm (r = 3.5 cm) and height 10 cm. Find its lateral (curved) surface area in cm^2.

Accepted Answer

Introduction / Context: Lateral surface area (LSA) for a right circular cone uses the formula πrl where l is the slant height. Given height (h) and radius (r), first compute l = √(r^2 + h^2). This problem checks Pythagorean use and then substitution into πrl. Given Data / Assumptions: Diameter = 7 cm → r = 3.5 cm h = 10 cm l = √(r^2 + h^2) LSA = πrl Concept / Approach: Compute slant height accurately; then multiply by πr. Rounding at the last step ensures agreement with the nearest option provided. Step-by-Step Solution: l = √(3.5^2 + 10^2) = √(12.25 + 100) = √112.25 ≈ 10.6 cmLSA ≈ π * 3.5 * 10.6 ≈ 37.1π ≈ 116.5 cm^2 (π ≈ 3.14)Using π = 22/7: LSA = (22/7) * 3.5 * 10.6 = 11 * 10.6 ≈ 116.6 cm^2Closest given option ≈ 110 cm^2 Verification / Alternative check: Small deviations arise from rounding l (true l ≈ 10.595). With exam rounding conventions, the listed nearest value is 110 cm^2. Why Other Options Are Wrong: 100, 70, 49 are too small; 120 slightly overestimates. 110 cm^2 is the closest to the computed ~116–117 cm^2 within typical rounding in such problems. Common Pitfalls: Using height instead of slant height in πrl; rounding too early on l leading to larger discrepancy. Final Answer: 110 sq cm

Question 17

Volume of a cone from radius and height: Find the volume of a cone with base radius 10 cm and height 21 cm. Express the answer approximately in cm^3.

Accepted Answer

Introduction / Context: The volume of a right circular cone is V = (1/3)πr^2h. Given r and h, this is a direct plug-in. Options are in approximate integers, so a decimal approximation with π works best here. Given Data / Assumptions: r = 10 cm h = 21 cm V = (1/3)πr^2h Concept / Approach: Compute r^2h first, then multiply by π/3. Using π ≈ 3.14 gives a tidy value near 2200 cm^3. Step-by-Step Solution: r^2h = 10^2 * 21 = 100 * 21 = 2100V = (1/3)π * 2100 = 700π ≈ 700 * 3.14 ≈ 2198 cm^3Rounded to the nearest option → 2200 cm^3 Verification / Alternative check: Using π = 22/7 gives V = 700 * (22/7) = 2200 cm^3 exactly, confirming the choice. Why Other Options Are Wrong: 3000, 5600, 6600 are too large given r and h; 2100 neglects the π/3 factor; 2200 is consistent with π = 22/7. Common Pitfalls: Using πr^2h (cylinder volume) instead of (1/3)πr^2h; arithmetic slips squaring r; rounding too early. Final Answer: 2200 cm^3

Question 18

Curved surface area of a cone (approximate): A cone has base diameter 6 cm (r = 3 cm) and altitude 4 cm. Find its curved surface area approximately (in cm^2).

Accepted Answer

Introduction / Context: The curved (lateral) surface area of a right circular cone equals πrl, where l is the slant height. Given r and vertical height h, compute l from the Pythagorean relation l = √(r^2 + h^2) and then substitute into πrl. This is a classic 3-4-5 triangle scenario. Given Data / Assumptions: Diameter = 6 cm → r = 3 cm Height h = 4 cm l = √(r^2 + h^2) = √(9 + 16) = √25 = 5 cm CSA = πrl Concept / Approach: Substitute r and l directly into πrl. Since options are approximate integers, evaluate with π ≈ 3.14 and round to the closest option. Step-by-Step Solution: CSA = π * 3 * 5 = 15π ≈ 47.1 cm^2Rounded → 47 cm^2 Verification / Alternative check: With π = 22/7, CSA = 15 * 22/7 ≈ 47.14, still rounds to 47 cm^2, confirming the choice. Why Other Options Are Wrong: 45 and 49 bracket but do not match the precise 47.1; 51 is too high; 43 is too low for l = 5 cm and r = 3 cm. Common Pitfalls: Using base area πr^2 or total surface area instead of curved area; miscomputing l (it is 5 by the 3-4-5 triangle). Final Answer: 47 cm2

Question 19

Cone – find slant height from volume and vertical height: A right circular cone has volume 100π cm^3 and vertical height 12 cm. Find its slant height l (in cm).

Accepted Answer

Introduction / Context: Given a cone’s volume and height, we can determine its base radius r via V = (1/3)πr^2h. With r and h known, the slant height l follows from the Pythagorean relation l = √(r^2 + h^2). This integrates volume and geometry of cones. Given Data / Assumptions: V = 100π cm^3 h = 12 cm V = (1/3)πr^2h l = √(r^2 + h^2) Concept / Approach: Solve (1/3)πr^2h = 100π → cancel π → r^2 = 300 / h. Then compute r and finally l using the right-triangle relation with h as one leg and r as the other. Step-by-Step Solution: (1/3)πr^2 * 12 = 100π → 4r^2 = 100 → r^2 = 25 → r = 5 cml = √(r^2 + h^2) = √(25 + 144) = √169 = 13 cm Verification / Alternative check: Check volume with r = 5, h = 12: V = (1/3)π * 25 * 12 = (1/3)π * 300 = 100π cm^3, matching the given value exactly. Why Other Options Are Wrong: 16 and 26 imply larger r or h; 9 is too small; 12 confuses vertical height with slant height. Only 13 cm satisfies both constraints simultaneously. Common Pitfalls: Forgetting to cancel π; using l = r + h incorrectly; rounding r before squaring; mixing slant and vertical height definitions. Final Answer: 13 cm

Question 20

Right circular cone — find base diameter from volume and height: The volume of a right circular cone is 48π cm3 and its height is 9 cm. Compute the diameter of the circular base (in centimeters).

Accepted Answer

Introduction / Context: This is a direct application of the cone volume formula with one unknown (the base radius). Once the radius is determined, the base diameter is twice the radius. Given Data / Assumptions: Height h = 9 cm. Volume V = 48π cm3. For a right circular cone, V = (1/3) * π * r^2 * h. All dimensions are in centimeters; π is symbolic (exact cancellation occurs). Concept / Approach: Rearrange the cone volume formula to isolate r^2. Then compute the diameter D = 2r. The calculation is purely algebraic and does not need any approximation to π since π cancels out. Step-by-Step Solution: V = (1/3) * π * r^2 * h48π = (1/3) * π * r^2 * 948π = 3π * r^2 ⇒ r^2 = 48/3 = 16r = 4 cm ⇒ D = 2r = 8 cm Verification / Alternative check: Substitute r = 4, h = 9 back into V: (1/3)*π*16*9 = (1/3)*π*144 = 48π cm3, exactly as given. Why Other Options Are Wrong: 4 cm is the radius, not diameter; 7 cm and 11 cm are arbitrary; they do not satisfy the cone volume with h = 9 cm. Common Pitfalls: Confusing radius with diameter; forgetting that r^2 appears in the formula; attempting to approximate π unnecessarily. Final Answer: 8 cm

Volume and Surface Area Questions