Multiple choice

Find the volume, in cubic cm, of the largest possible cylinder that can be carved out of a sphere whose radius is 2√3 cm.

  1. 32π

  2. 16π

  3. 48π

  4. None of the above
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A Correct answer
Explanation

For a cylinder of radius r and height h in a sphere of radius R, r^2 + (h/2)^2 = R^2. Volume V = pi * r^2 * h = pi * (R^2 - h^2/4) * h = pi * (12h - h^3/4). Maximize: dV/dh = pi * (12 - 3h^2/4) = 0 => h^2 = 16 => h = 4. r^2 = 12 - 4 = 8. V = pi * 8 * 4 = 32 * pi.

AI explanation

Using the relationship between the sphere's radius, the cylinder's radius (r), and its height (2h) given by r squared plus h squared equals 12, we apply the derivative method to maximize the cylinder's volume. Expressing the volume formula as V = pi times (12 minus h squared) times h and setting its derivative to zero yields h equals 2. Substituting r as the square root of 8 and height as 4 into the volume formula gives the maximum volume of 32 pi.

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