(1) The biggest cone that fits inside a cube of side 2 cm has a base diameter and height both equal to 2 cm. Radius = 1 cm, height = 2 cm.
Volume = (1/3)πr²h = (1/3)π(1)²(2) = 2π/3 cm³.
(2) Original curved surface area = 2πrh.
New radius = r/2, new height = 2h.
New curved surface area = 2π(r/2)(2h) = 2πrh.
The area remains the same, so the ratio is 1:1.
(3) Volume of cube = 12 cm³. Side of cube = ∛12 cm.
Diameter of sphere = side of cube = ∛12 cm. Radius = (∛12)/2 cm.
Volume of sphere = (4/3)πr³ = (4/3)π[(∛12)/2]³ = (4/3)π(12/8) = (4/3)π(3/2) = 2π cm³.
(4) Volume of cone = (1/3)πr²h = (1/3)π(14)²(7) = (1/3)π(196)(7) = (1372/3)π cm³.
Volume of sphere = (4/3)πR³.
Set equal: (4/3)πR³ = (1372/3)π → 4R³ = 1372 → R³ = 343 → R = 7 cm.
(5) Biggest sphere in cube of side 8r has diameter = 8r, so radius = 4r.
Volume = (4/3)π(4r)³ = (4/3)π(64r³) = 256πr³/3.
(6) Surface area increases by factor of 4. Since surface area ∝ r², r increases by factor of √4 = 2.
Change in radius: new radius = 2 × original radius. Change = 2r - r = r (i.e., it doubles).
(7) Biggest sphere in cube of side 6r has diameter = 6r, so radius = 3r.
Surface area = 4πr² = 4π(3r)² = 4π(9r²) = 36πr².
(8) Radius = 4a. Surface area = 4πr² = 4π(4a)² = 4π(16a²) = 64πa².
(9) Cone volume = (1/3)πr²h, Hemisphere volume = (2/3)πr³. Given h = r.
Cone volume = (1/3)πr³, Hemisphere volume = (2/3)πr³.
Ratio (Cone : Hemisphere) = [(1/3)πr³] / [(2/3)πr³] = 1/2.
Answer: c. 1:2
(10) Curved surface area of hemisphere = 2πr². Given r = 3r (using variable r).
Area = 2π(3r)² = 2π(9r²) = 18πr².
Answer: d. 18 π r²
(11) Volume of sphere = (4/3)πr³. Given r = 3a.
Volume = (4/3)π(3a)³ = (4/3)π(27a³) = 36πa³.
Answer: d. 36 π a³
Parent Tip: Review the logic above to help your child master the concept of surface area and volume practice worksheet.
