Question 1

Conical caps — sheet required for multiple identical cones: Shantanu’s party cap is a right circular cone with base radius 7 cm and height 24 cm. Find the total sheet area required to make 5 such caps (assume each cap uses only curved surface area a…

Accepted Answer

Introduction / Context: Paper or thin-sheet conical caps are made from the lateral (curved) surface of a right circular cone. The needed sheet equals the curved surface area (CSA) of a cone, not the total surface area (which includes the base). For multiple identical caps, multiply the one-cap CSA by the number of caps. Given Data / Assumptions: Radius r = 7 cm. Height h = 24 cm. Number of caps n = 5. Sheet used = lateral surface only (CSA). Concept / Approach: Compute slant height l using the Pythagorean relation l = sqrt(r^2 + h^2). Then CSA per cone is π * r * l. Multiply by 5 for five caps. Round sensibly to match the nearest listed option if required. Step-by-Step Solution: l = sqrt(7^2 + 24^2) = sqrt(49 + 576) = sqrt(625) = 25 cmCSA (one cap) = π * r * l = π * 7 * 25 = 175π cm2Five caps ⇒ 5 * 175π = 875π ≈ 875 * 3.1416 ≈ 2741.6 cm2Nearest option ≈ 2750 sq cm Verification / Alternative check: Using π = 22/7 gives 875 * 22/7 = 2750 cm2 exactly, matching the option. Why Other Options Are Wrong: 2700 and 3000 are coarse approximations not aligned with π = 22/7; 5000 is far too high for these dimensions. Common Pitfalls: Including the base area (πr^2) mistakenly, or using height instead of slant height in the CSA formula. Final Answer: 2750 sq cm

Question 2

Keep cone volume fixed when base radius doubles — find new height: For a right circular cone, if the base radius is doubled and the volume is to remain unchanged, what must be the new height as a fraction of the previous height?

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Introduction / Context: The cone volume depends on the square of the radius and linearly on the height. If volume is to remain unchanged while radius changes, the height must adjust to compensate for the squared effect of radius. Given Data / Assumptions: Original volume V = (1/3)π r^2 h. New radius r′ = 2r. Volume constant: V′ = V. Concept / Approach: Since V ∝ r^2 h, keeping V fixed implies r^2 h = constant. If r becomes 2r, then r^2 grows by a factor of 4. Therefore, height must shrink by the same factor (1/4) so that the product r^2 h remains the same. Step-by-Step Solution: V′ = (1/3)π (2r)^2 h′ = (1/3)π * 4r^2 * h′Set V′ = V = (1/3)π r^2 h ⇒ 4r^2 h′ = r^2 hh′ = h / 4 Verification / Alternative check: Insert any numbers (e.g., r = 3, h = 12): V ∝ 9 * 12 = 108. With r′ = 6, h′ should be 3: 36 * 3 = 108, same product. Why Other Options Are Wrong: Half compensates for a linear change, not quadratic; one-third and 1/√2 do not neutralize the 4× increase from radius doubling. Common Pitfalls: Forgetting that radius is squared in the formula; trying to keep height inversely proportional to r rather than r^2. Final Answer: one-fourth of the previous height

Question 3

Cone with known diameter and slant height — compute volume via height: A right circular cone has diameter 14 m (radius 7 m) and slant height 9 m. Find its volume (in m3).

Accepted Answer

Introduction / Context: To compute a cone’s volume when radius and slant height are given, first find the vertical height using Pythagoras on the right triangle formed by radius, height, and slant height. Then apply the standard cone volume formula. Given Data / Assumptions: Radius r = 7 m (diameter 14 m). Slant height l = 9 m. Height h satisfies l^2 = r^2 + h^2. Concept / Approach: Compute h = sqrt(l^2 − r^2). Then volume V = (1/3)π r^2 h. Leave the radical in simplest exact form for a clean expression. Step-by-Step Solution: h = sqrt(9^2 − 7^2) = sqrt(81 − 49) = sqrt(32)V = (1/3)π * 7^2 * sqrt(32) = (1/3)π * 49 * sqrt(32)Therefore, V = 49 π √32 / 3 m3 Verification / Alternative check: Approximate: √32 ≈ 5.657; V ≈ (49 * 5.657 / 3)π ≈ (277.2/3)π ≈ 92.4π m3, consistent. Why Other Options Are Wrong: All other forms mis-handle constants or the radical; the correct combination of r^2 and √32 is specific. Common Pitfalls: Using l in place of h in the volume formula; forgetting to square the radius. Final Answer: 49 π √32/3 m3

Question 4

Two cones — deduce height ratio from volume and radius ratios: The volumes of two cones are in the ratio 2 : 3 and the radii of their bases are in the ratio 1 : 2. Find the ratio of their heights.

Accepted Answer

Introduction / Context: For cones, volume is proportional to r^2 h. Given the ratio of volumes and radii, one can solve for the ratio of heights using proportional reasoning, without computing any absolute values. Given Data / Assumptions: V1 : V2 = 2 : 3. r1 : r2 = 1 : 2. V ∝ r^2 h for fixed proportionality constant (1/3)π. Concept / Approach: Set (r1^2 h1) : (r2^2 h2) = 2 : 3. Substitute r1^2 : r2^2 = 1^2 : 2^2 = 1 : 4 and solve for h1 : h2. Step-by-Step Solution: (1 * h1) : (4 * h2) = 2 : 3Cross-multiply: 3h1 = 8h2 ⇒ h1 : h2 = 8 : 3 Verification / Alternative check: Try concrete numbers, e.g., r1 = 1, r2 = 2; pick h1 = 8, h2 = 3. Then V1 ∝ 1*8=8 and V2 ∝ 4*3=12 → 8:12 = 2:3 as required. Why Other Options Are Wrong: They either invert the ratio or ignore the square on radius in the volume expression. Common Pitfalls: Using r instead of r^2 when connecting volume ratio to height ratio. Final Answer: 8 : 3

Question 5

Cone with r : h = 5 : 12 and known volume — find slant height: A right circular cone has radius : height = 5 : 12 and volume 314 2/7 cm3. Find its slant height (in centimeters).

Accepted Answer

Introduction / Context: When a cone’s radius and height are in a fixed ratio, we can parameterize them with a common factor and use the volume to determine that factor. The slant height then follows from Pythagoras. Given Data / Assumptions: r : h = 5 : 12 ⇒ r = 5k, h = 12k for some k > 0. V = 314 2/7 cm3 = 2200/7 cm3 (since 314*7 + 2 = 2200). Use π = 22/7 (implied by mixed fraction format), typical in aptitude problems. Concept / Approach: Compute V in terms of k: V = (1/3)π r^2 h = (1/3)π * (25k^2) * (12k) = 100π k^3. Equate to 2200/7 to obtain k, then compute slant height l = sqrt(r^2 + h^2). Step-by-Step Solution: 100π k^3 = 2200/7 with π = 22/7 ⇒ 100 * (22/7) * k^3 = 2200/7Cancel (22/7): 100k^3 = 100 ⇒ k = 1Thus r = 5 cm, h = 12 cml = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 cm Verification / Alternative check: Plug r = 5, h = 12 back into V: (1/3)*π*25*12 = 100π = 100*(22/7) = 2200/7 cm3, matching the data. Why Other Options Are Wrong: 15, 16, 18 cm do not satisfy l^2 = r^2 + h^2 with r : h = 5 : 12. Common Pitfalls: Mistaking the mixed fraction as 314.27; forgetting to square r in the volume formula. Final Answer: 13 cm

Question 6

Cone volume change — height tripled and radius halved: A right circular cone’s height is increased by 200% (so it becomes three times), while its base radius is reduced by 50%. What is the resulting change in the cone’s volume?

Accepted Answer

Introduction / Context: The cone volume V = (1/3)π r^2 h scales with r^2 and h. When r and h change by given factors, the new volume is obtained by multiplying the old volume by the product of those factors, respecting the square on r. Given Data / Assumptions: h becomes 3h (200% increase means h → h + 2h = 3h). r becomes 0.5r (50% reduction). All else unchanged. Concept / Approach: New/old factor f = (r′/r)^2 * (h′/h) = (0.5)^2 * 3 = 0.25 * 3 = 0.75. Therefore, V decreases by 25%. Step-by-Step Solution: V′ = (1/3)π (0.5r)^2 (3h) = (1/3)π * 0.25 r^2 * 3h = 0.75 * VPercent change = (0.75 − 1) * 100% = −25% Verification / Alternative check: Example: r = 2, h = 2 ⇒ V = (1/3)π*4*2 = (8/3)π; new r = 1, new h = 6 ⇒ V′ = (1/3)π*1*6 = 2π; ratio 2π : (8/3)π = 0.75. Why Other Options Are Wrong: They ignore the squared dependence on r or misinterpret “200% increase.” Common Pitfalls: Adding percentage changes rather than multiplying the correct factors. Final Answer: decreases by 25%

Question 7

Two cones — volumes 1 : 3 and diameters 3 : 5 — find heights ratio: The volumes of two right circular cones are in the ratio 1 : 3 and their diameters are in the ratio 3 : 5. Find the ratio of their heights.

Accepted Answer

Introduction / Context: Using proportionality V ∝ r^2 h, a known ratio of volumes and radii (or diameters) lets us solve for the ratio of heights. Since diameter and radius scale the same, we can use the diameter ratio directly to infer the radius ratio. Given Data / Assumptions: V1 : V2 = 1 : 3. D1 : D2 = 3 : 5 ⇒ r1 : r2 = 3 : 5. Concept / Approach: Set (r1^2 h1) : (r2^2 h2) = 1 : 3. Substitute r1^2 : r2^2 = 9 : 25, then solve for h1 : h2. Step-by-Step Solution: (9 h1) : (25 h2) = 1 : 3Cross-multiply: 3 * 9 h1 = 25 h2 ⇒ h1 : h2 = 25 : 27 Verification / Alternative check: Choose r1 = 3, r2 = 5 and heights in 25 : 27; then V1 : V2 ∝ (9*25) : (25*27) = 225 : 675 = 1 : 3. Why Other Options Are Wrong: They ignore the square on r or put the ratios in the wrong order. Common Pitfalls: Using r instead of r^2 in the proportional relation to volume. Final Answer: 25 : 27

Question 8

Cones with equal diameters — relate curved surface areas via slant heights: Two cones have equal diameters (thus equal radii). If their slant heights are in the ratio 5 : 7, find the ratio of their curved surface areas.

Accepted Answer

Introduction / Context: The curved surface area (CSA) of a cone equals π r l. If two cones have the same radius, then CSA varies directly with slant height l. Therefore, the ratio of CSAs equals the ratio of slant heights. Given Data / Assumptions: Equal diameters ⇒ r1 = r2. l1 : l2 = 5 : 7. Concept / Approach: CSA1 : CSA2 = (π r l1) : (π r l2) = l1 : l2 when r is common. No additional computation is required. Step-by-Step Solution: CSA ratio = l1 : l2 = 5 : 7 Verification / Alternative check: Pick r = 10, l1 = 5, l2 = 7: CSAs are 50π and 70π, giving 5 : 7. Why Other Options Are Wrong: 25 : 49 is the ratio of l^2 and would apply to area only if area depended on l^2 (it does not here since r is constant). Common Pitfalls: Confusing cone CSA (π r l) with total surface area (π r l + π r^2), which would still keep a linear l dependence when r is common. Final Answer: 5 : 7

Question 9

Cone with fixed height — radius increased by 15%: effect on volume: For a right circular cone, the base radius increases by 15% while the height remains fixed. By what percentage does the volume increase?

Accepted Answer

Introduction / Context: With height fixed, a cone’s volume is proportional to r^2. A percentage increase in radius thus compounds as the square of the radius factor for the volume. Given Data / Assumptions: r → 1.15 r. h unchanged. V ∝ r^2 when h is fixed. Concept / Approach: Compute the volume scale factor as (1.15)^2. Convert that factor into a percentage increase by subtracting 1 and multiplying by 100%. Step-by-Step Solution: New factor = (1.15)^2 = 1.3225Percent increase = (1.3225 − 1)*100% = 32.25% Verification / Alternative check: For small p, the approximation would be ≈ 2p = 30%, but the exact square gives 32.25%, slightly higher. Why Other Options Are Wrong: 30% is a linear approximation; 31% and 34.75% are off the exact square; 32.25% matches the precise compounding. Common Pitfalls: Applying the linear percentage directly to volume rather than squaring the radius factor. Final Answer: 32.25%

Question 10

Largest circle on a sphere — compute diameter from sphere surface area: The surface area of a sphere is 616 sq cm. What is the diameter of the largest circle lying on its surface (i.e., the great circle)?

Accepted Answer

Introduction / Context: The largest circle on a sphere is a great circle whose diameter equals the sphere’s diameter. Therefore, determining the sphere’s radius from its surface area directly gives the required diameter. Given Data / Assumptions: Sphere surface area S = 4πr^2 = 616 cm2. Standard school setting typically uses π = 22/7 for such integer-friendly values. Concept / Approach: Solve for r^2 = 616 / (4π). With π = 22/7, arithmetic becomes exact and yields an integer r. The great-circle diameter equals 2r. Step-by-Step Solution: r^2 = 616 / (4π) = 154 / πUsing π = 22/7 ⇒ r^2 = 154 * 7 / 22 = 49r = 7 cm ⇒ diameter = 2r = 14 cm Verification / Alternative check: Back substitute: 4πr^2 = 4*(22/7)*49 = 616 cm2, matching. Why Other Options Are Wrong: 10.5 cm is the radius, not the diameter; 7 cm and 3.5 cm are radius and half-radius magnitudes. Common Pitfalls: Confusing surface area with cross-sectional (circle) area; forgetting the great circle’s diameter equals the sphere’s diameter. Final Answer: 14 cm

Question 11

Moon vs Earth — diameter ratio 1 : 4 — find volume ratio: The Moon’s diameter is approximately one-fourth of Earth’s diameter. What is the approximate ratio of their volumes (Moon : Earth)?

Accepted Answer

Introduction / Context: Solid volumes scale with the cube of the linear dimension. If one sphere’s diameter is a fixed fraction of another’s, their volumes are in the cube of that fraction. Given Data / Assumptions: d_Moon : d_Earth = 1 : 4. Volumes of spheres V ∝ d^3 (equivalently r^3). Concept / Approach: Compute (1/4)^3 = 1/64 to get Moon : Earth volume ratio. Step-by-Step Solution: V_Moon : V_Earth = (1^3) : (4^3) = 1 : 64 Verification / Alternative check: Using radii gives the same result since radius is proportional to diameter. Why Other Options Are Wrong: 1 : 4 and 4 : 16 ignore cubic scaling; 1 : 128 would correspond to diameter ratio 1 : 5 approximately, not 1 : 4. Common Pitfalls: Applying linear or square scaling to a volume comparison. Final Answer: 1 : 64

Question 12

Sphere vs hemisphere with equal total surface area — compare volumes: A sphere and a hemisphere have the same total surface area (hemisphere area includes its circular base). Find the ratio of their volumes (sphere : hemisphere).

Accepted Answer

Introduction / Context: Be careful with the hemisphere’s surface area: “total” means curved area plus the flat circular base. Setting the total areas equal lets us link their radii and then compare volumes. Given Data / Assumptions: Sphere: S_s = 4πR^2, V_s = (4/3)πR^3. Hemisphere (including base): S_h = 2πr^2 + πr^2 = 3πr^2, V_h = (2/3)πr^3. Equal total surface areas: 4πR^2 = 3πr^2. Concept / Approach: From 4πR^2 = 3πr^2, deduce r^2 = (4/3)R^2 ⇒ r = (2/√3)R. Substitute into the volume expressions and simplify the ratio V_s : V_h. Step-by-Step Solution: Set 4πR^2 = 3πr^2 ⇒ r = (2/√3)RV_s : V_h = [(4/3)πR^3] : [(2/3)πr^3] = 2 * (R^3 / r^3)R/r = √3/2 ⇒ (R/r)^3 = (√3/2)^3 = 3√3/8Thus V_s : V_h = 2 * (3√3/8) = 3√3/4 : 1 Verification / Alternative check: Pick R = 2 and compute r = 2/√3; substituting numerically reproduces the symbolic ratio. Why Other Options Are Wrong: They drop or misplace the factor 2 or the cube when converting area equality into a volume ratio. Common Pitfalls: Using hemisphere curved area only (2πr^2) instead of total (3πr^2). Final Answer: 3√3/4 : 1

Question 13

Lead balls from a larger sphere — count by volume ratio: How many solid lead balls of diameter 2 cm can be made by melting a sphere of diameter 16 cm (no loss of material)?

Accepted Answer

Introduction / Context: When a large sphere is recast into smaller spheres, the number produced equals the ratio of the large volume to the small volume. Because sphere volume is proportional to the cube of radius (or diameter), the count reduces to a simple cube ratio of diameters. Given Data / Assumptions: Large sphere diameter D_L = 16 cm ⇒ radius R_L = 8 cm. Small ball diameter D_s = 2 cm ⇒ radius r_s = 1 cm. No wastage or gaps; exact volume conservation. Concept / Approach: Number N = V_L / V_s = ( (4/3)πR_L^3 ) / ( (4/3)πr_s^3 ) = (R_L / r_s)^3 = (8/1)^3. Step-by-Step Solution: N = (8)^3 = 512 Verification / Alternative check: Compute actual volumes: V_L = (4/3)π*512 = (2048/3)π; V_s = (4/3)π*1 = (4/3)π; their ratio is 512. Why Other Options Are Wrong: 2048, 2055, 2058 are far larger than the correct cube ratio; they appear to confuse volume with surface area or linear scaling. Common Pitfalls: Using diameter squared instead of cubed; forgetting to convert diameters to radii consistently (though it cancels here). Final Answer: 512 (not listed) ⇒ None of the above

Question 14

Hemispherical bowl painted inside and outside — compute total cost: A hemispherical bowl of radius 3.5 cm is to be painted on the inside as well as the outside. If the rate is ₹ 5 per 10 sq cm, find the total painting cost.

Accepted Answer

Introduction / Context: The paint is applied to both the inner and outer curved surfaces of a thin hemispherical shell. Neglecting thickness, the inner and outer radii are effectively the same, so the total painted area is twice the curved area of a hemisphere. Given Data / Assumptions: Radius r = 3.5 cm. Curved surface area (hemisphere) = 2πr^2. Painting both sides ⇒ total curved area = 2 * (2πr^2) = 4πr^2. Rate = ₹ 5 per 10 cm2 = ₹ 0.5 per cm2. Concept / Approach: Compute total area 4πr^2 and multiply by ₹ 0.5 per cm2. Using π = 22/7 makes the arithmetic exact for r = 3.5. Step-by-Step Solution: 4πr^2 = 4π * (3.5)^2 = 4π * 12.25 = 49π cm2With π = 22/7 ⇒ 49π = 49 * 22/7 = 154 cm2Cost = 0.5 * 154 = ₹ 77 Verification / Alternative check: Compute each curved side separately: 2πr^2 ≈ 76.97 cm2; doubled ≈ 153.94 cm2; times ₹ 0.5 ≈ ₹ 76.97 ≈ ₹ 77. Why Other Options Are Wrong: ₹ 50, ₹ 56, ₹ 81 do not match the exact evaluation with π = 22/7. Common Pitfalls: Accidentally adding the flat circular rim area; using π = 3.14 with premature rounding may slightly change cents but not the integer rupee choice. Final Answer: ₹ 77

Question 15

Sphere with given surface area — compute volume exactly: If a sphere has surface area 616 sq cm, find its volume (in cubic centimeters).

Accepted Answer

Introduction / Context: Given a sphere’s surface area, solve for its radius and then compute the volume. The numbers are chosen so that using π = 22/7 leads to exact integer simplification. Given Data / Assumptions: S = 4πr^2 = 616 cm2. π = 22/7 (standard in such aptitude items). Volume V = (4/3)πr^3. Concept / Approach: From 4πr^2 = 616, obtain r first. Then substitute into V = (4/3)πr^3 and simplify exactly. Step-by-Step Solution: r^2 = 616/(4π) = 154/π = 154 * 7/22 = 49 ⇒ r = 7 cmV = (4/3)πr^3 = (4/3)π * 343 = (1372/3)πWith π = 22/7: V = (1372/3) * (22/7) = (196 * 22)/3 = 4312/3 cm3 Verification / Alternative check: Decimal check: 4312/3 ≈ 1437.33 cm3, reasonable for r = 7 cm. Why Other Options Are Wrong: Other fractions do not match the exact cancellation with r = 7 and π = 22/7. Common Pitfalls: Using 616/π as if it were 616/3.14 with premature rounding; mis-evaluating r from S. Final Answer: 4312/3 cm3

Question 16

Two spheres — ratio of diameters 3 : 5 — find ratio of surface areas: If the diameters of two spheres are in the ratio 3 : 5, what is the ratio of their surface areas?

Accepted Answer

Introduction / Context: Surface area of a sphere depends on the square of its radius (or diameter). Thus, a ratio of diameters translates to the square of that ratio for surface areas. Given Data / Assumptions: d1 : d2 = 3 : 5 ⇒ r1 : r2 = 3 : 5. S ∝ r^2 (or ∝ d^2). Concept / Approach: Square the linear ratio to obtain the surface area ratio. Step-by-Step Solution: S1 : S2 = (3^2) : (5^2) = 9 : 25 Verification / Alternative check: Pick r1 = 3, r2 = 5; S1 = 36π, S2 = 100π ⇒ 36π : 100π = 9 : 25. Why Other Options Are Wrong: 9 : 10 and 3 : 5 are linear ratios; 27 : 125 applies to volumes (cube). Common Pitfalls: Confusing surface-area scaling (square) with volume scaling (cube). Final Answer: 9 : 25

Question 17

Large sphere split into 64 equal small spheres — find each small sphere’s surface area: A big sphere of diameter 8 cm is divided into 64 identical small spheres. What is the surface area of one small sphere?

Accepted Answer

Introduction / Context: Equal-volume partition implies that the volume of the large sphere equals 64 times the volume of each small sphere. Use this to derive each small sphere’s radius, then compute its surface area. Given Data / Assumptions: Large diameter = 8 cm ⇒ large radius R = 4 cm. Number of small spheres = 64. Volume conservation: (4/3)πR^3 = 64 * (4/3)πr^3. Concept / Approach: Cancel constants and solve for r. Then S_small = 4πr^2. Step-by-Step Solution: R^3 = 64 r^3 ⇒ r^3 = R^3 / 64 = 64 / 64 = 1 ⇒ r = 1 cmS_small = 4πr^2 = 4π * 1^2 = 4π cm2 Verification / Alternative check: Total surface area is not conserved—only volume is. The question asks for one small sphere, which is 4π cm2. Why Other Options Are Wrong: π and 2π are too small; 8π is double the correct area. Common Pitfalls: Assuming surface area is conserved during partitioning; it is not. Final Answer: 4 π cm2

Question 18

Curved surface area of a hemisphere — diameter given: Find the curved surface area (CSA) of a hemisphere of diameter 28 cm.

Accepted Answer

Introduction / Context: The curved surface area of a hemisphere equals 2πr^2. With a diameter given, first compute the radius and then evaluate the expression, preferably using π = 22/7 for exact integer results with multiples of 7. Given Data / Assumptions: Diameter = 28 cm ⇒ r = 14 cm. CSA (hemisphere) = 2πr^2. Concept / Approach: Substitute r into 2πr^2 and simplify exactly. Step-by-Step Solution: CSA = 2π * 14^2 = 2π * 196 = 392πWith π = 22/7: 392π = 392 * 22/7 = 56 * 22 = 1232 cm2 Verification / Alternative check: Using π ≈ 3.1416 gives ≈ 1231.5 cm2, rounding to the nearest integer 1232. Why Other Options Are Wrong: 1236, 1238, 1233 are near but not equal to the exact evaluation. Common Pitfalls: Confusing hemisphere CSA (2πr^2) with total surface area (3πr^2). Final Answer: 1232 sq cm

Question 19

Hemisphere of diameter 2 cm — difference (TSA − CSA): For a hemisphere with diameter 2 cm (radius 1 cm), find the difference between its total surface area and its curved surface area.

Accepted Answer

Introduction / Context: For a hemisphere, the total surface area (TSA) includes the curved area plus the area of the flat circular base, while the curved surface area (CSA) excludes the base. Their difference is exactly the base area. Given Data / Assumptions: Diameter = 2 cm ⇒ r = 1 cm. TSA (hemisphere) = 3πr^2. CSA (hemisphere) = 2πr^2. Concept / Approach: Compute TSA − CSA = (3πr^2 − 2πr^2) = πr^2 = π for r = 1. Step-by-Step Solution: TSA − CSA = πr^2 = π * 1^2 = π cm2 Verification / Alternative check: Geometrically, this is just the area of the base circle, which must be πr^2. Why Other Options Are Wrong: 2π, 3π, 4π correspond to 2, 3, 4 times the base area and do not apply to the difference. Common Pitfalls: Using diameter instead of radius in the circle area formula; mixing TSA with CSA expressions. Final Answer: π sq cm

Question 20

A solid metallic sphere of radius 12 cm is melted and recast into three smaller solid spheres. If two of the new spheres have radii 6 cm and 8 cm respectively, what is the radius (in cm) of the third sphere?

Accepted Answer

Introduction / Context: When a solid is melted and recast, total volume is conserved (no loss of material). Here, one large sphere is melted into three smaller spheres. Using the sphere volume formula, we equate the original volume to the sum of the three new volumes and solve for the unknown radius. Given Data / Assumptions: Original radius R = 12 cm. Two new radii r1 = 6 cm and r2 = 8 cm. Third radius r3 = ? Use volume of sphere: V = (4/3) * π * r^3. Concept / Approach: Volume conservation: (4/3)πR^3 = (4/3)πr1^3 + (4/3)πr2^3 + (4/3)πr3^3 ⇒ R^3 = r1^3 + r2^3 + r3^3. Cancel (4/3)π from both sides. Step-by-Step Solution: R^3 = 12^3 = 1728r1^3 = 6^3 = 216r2^3 = 8^3 = 512r3^3 = R^3 − r1^3 − r2^3 = 1728 − 216 − 512 = 1000r3 = 1000^(1/3) = 10 cm Verification / Alternative check: Compute exact volumes: (4/3)π(1728) on LHS and (4/3)π(216 + 512 + 1000) on RHS. The sums match exactly, confirming r3 = 10 cm. Why Other Options Are Wrong: 12 cm: Would imply zero melting into smaller spheres; contradicts given data. 14 cm or 16 cm: Their cubes greatly exceed the remaining volume after subtracting the first two spheres. Common Pitfalls: Forgetting volume scales with the cube of radius; using surface area instead of volume; arithmetic slips when summing cubes. Final Answer: 10 cm

Volume and Surface Area Questions