Multiple choice

Two right circular cones having same base radius of $2.1$ cm but heights $4.1$ cm and $4.3$ cm are melted and recast into a single sphere Find the radius of the sphere so formed

  1. $2.1$ cm
  2. $2$ cm
  3. $4.2$ cm
  4. $3.1$ cm
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A Correct answer
Explanation

Volume of sphere = sum of volumes of two cones. (4/3)piR^3 = (1/3)pir^2h1 + (1/3)pir^2h2. (4/3)R^3 = (1/3)r^2(h1+h2). 4R^3 = (2.1)^2 * (4.1+4.3) = 4.41 * 8.4 = 37.044. R^3 = 37.044 / 4 = 9.261. R = cube root of 9.261 = 2.1.

AI explanation

The combined volume of the two cones equals the volume of the new sphere, using the formulas one third times pi times r squared times h and four thirds times pi times r cubed. The total volume of the two cones is one third times pi times 2.1 squared times 4.1 plus one third times pi times 2.1 squared times 4.3. This simplifies to one third times pi times 4.41 times 8.4, which equals 12.348 times pi. Setting this equal to four thirds times pi times the new radius cubed and solving gives the radius of the sphere as 2.1 cm.

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