Multiple choice

The total surface area of a metallic hemisphere is $1848\ cm^{2}$. The hemisphere is melted to form a solid right circular cone. If the radius of the base of the cone is the same as the radius of the hemisphere, its height is

  1. $42\ cm$
  2. $26\ cm$
  3. $28\ cm$
  4. $30\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

TSA of hemisphere = 3 * pi * r^2 = 1848. r^2 = 1848 / (3 * 22/7) = 1848 * 7 / 66 = 196. r = 14. Volume of hemisphere = 2/3 * pi * r^3 = 2/3 * 22/7 * 14^3 = 5749.33. Volume of cone = 1/3 * pi * r^2 * h = 1/3 * 22/7 * 14^2 * h = 5749.33. 1/3 * 22/7 * 196 * h = 5749.33. h = 28.

AI explanation

Using the total surface area formula for a hemisphere, 3*pi*r^2 equals 1848, so pi*r^2 equals 616. Because the volumes remain equal during melting, the hemisphere volume formula (2/3)pi*r^3 equals the cone volume formula (1/3)*pi*r^2*h. Since the cone base radius is r, the height h calculates to 2*r. The hemisphere radius is found as 1848 divided by (3(22/7)) equals 14 cm, making the required height 28 cm. The result is 28 cm.

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