Multiple choice

Three solid spheres of radius $1$ cm, $6$ cm and $8$ cm, respectively are melted together and cast into a single sphere. The radius $r$ of this sphere will be

  1. $\displaystyle r< 9$ cm
  2. $r=9$ cm
  3. $\displaystyle 9$ cm $< r< 10$ cm
  4. $\displaystyle r> 10$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The volume of the new sphere equals the sum of the volumes of the three smaller spheres: (4/3)pi*r^3 = (4/3)pi*(1^3 + 6^3 + 8^3). Thus, r^3 = 1 + 216 + 512 = 729. Since 9^3 = 729, r = 9 cm.

AI explanation

By the conservation of volume, the volume of the new sphere equals the sum of the volumes of the three original spheres, calculated as 4/3 * pi * (1^3 + 6^3 + 8^3). Adding the cubes gives 1 + 216 + 512 = 729, so the new sphere has a volume of 4/3 * pi * 729. Equating this to the volume formula 4/3 * pi * r^3 means r^3 = 729, and taking the cube root reveals that r = 9 cm.

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