Surface Areas and Volumes — Practice

Question 1

Find volume of cylinder with radius 7 cm and height 10 cm.

Accepted Answer

✦ Answer: V = πr²h = (22/7)(49)(10) = 1540 cm³.

Question 2

Find slant height, CSA, and volume of cone with radius 6 cm and height 8 cm.

Accepted Answer

Step 1 — Slant height.
  l = √(r² + h²) = √(36 + 64) = √100 = 10 cm

Step 2 — CSA.
  CSA = πrl = (22/7)(6)(10) = 1320/7 ≈ 188.57 cm²

Step 3 — Volume.
  V = (1/3)πr²h = (1/3)(22/7)(36)(8) = 6336/21 = 301.71 cm³

✦ Answer: l = 10 cm, CSA ≈ 188.57 cm², V ≈ 301.71 cm³.

Question 3

A solid sphere of radius 10.5 cm is melted and recast into smaller cones of radius 3.5 cm and height 3 cm. Find the number of cones.

Accepted Answer

Step 1 — Volume of sphere.
  V_sphere = (4/3)π(10.5)³
  10.5³ = 1157.625
  V = (4/3)(22/7)(1157.625) = (88/21)(1157.625) ≈ 4851 cm³

Step 2 — Volume of one cone.
  V_cone = (1/3)π(3.5)²(3)
  = (1/3)(22/7)(12.25)(3)
  = (1/3)(22/7)(36.75)
  = 808.5/7 = 115.5 cm³
  Or: V_cone = (1/3)π × 12.25 × 3 = π × 12.25 = (22/7)(12.25) = 269.5/7 = 38.5 cm³
  Wait, let me recompute: (1/3)(22/7)(12.25)(3) = (22/7)(12.25) = 269.5/7 = 38.5 cm³ ✓

Step 3 — Number of cones.
  N = V_sphere / V_cone = 4851 / 38.5 = 126

Step 4 — Verify.
  126 × 38.5 = 4851 ✓

✦ Answer: Number of cones = 126.

Practice — Surface Areas and Volumes

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