A cone has height 8.4 cm (half of 16.8 cm) and base diameter 4.2 cm. It is melted and recast into a sphere. Find the surface area of the sphere (use π = 22/7).
Aptitude
Volume and Surface Area
Difficulty: Medium
Choose an option
-
A55.44 sq cm
-
B60 sq cm
-
C65.58 sq cm
-
D59 sq cm
-
ENone of these
Answer
Correct Answer: 55.44 sq cm
Explanation
Introduction / Context:We convert a cone into a sphere by melting, so volumes remain equal. From the equal volume we get the sphere radius, then compute the sphere’s surface area.
Given Data / Assumptions:
- Cone height h = 8.4 cm.
- Cone base diameter = 4.2 cm ⇒ radius r = 2.1 cm.
- π = 22/7.
- No loss of material: V_cone = V_sphere.
Concept / Approach:
- V_cone = (1/3)*π*r^2*h.
- V_sphere = (4/3)*π*R^3 ⇒ R from volume equality.
- Surface area S = 4*π*R^2.
Step-by-Step Solution:
V_cone = (1/3)*(22/7)*(2.1)^2*(8.4)(2.1)^2 = 4.41 ⇒ V_cone = (1/3)*(22/7)*4.41*8.4 ≈ 38.79 cm^3Set (4/3)*π*R^3 = ≈ 38.79 ⇒ R^3 ≈ 38.79 * 3 / (4π)Using π = 22/7 ⇒ R^3 ≈ 37.044 / 4 = 9.261 ⇒ R ≈ 2.09 cmSurface area S = 4*π*R^2 = 4*(22/7)*(2.09)^2 ≈ 17.47*π ≈ 55.4 cm^2Verification / Alternative check:Using more precise intermediate rounding still yields ≈ 55.4 cm^2, matching 55.44 sq cm among options.
Why Other Options Are Wrong:
- 60/65.58/59 sq cm: Deviate from the precise conversion steps.
- None of these: Not applicable since 55.44 sq cm fits.
Common Pitfalls:
- Using diameter instead of radius.
- Rounding too early when finding R then squaring.
Final Answer:55.44 sq cm