Multiple choice

Figure shows a semicircle that is the graph of the equation $y = \sqrt {6x - x^{2}}$. If the semicircle is rotated $360^{\circ}$ about the $x$-axis, calculate the volume of the sphere that is created.

  1. $6\pi$
  2. $12\pi$
  3. $18\pi$
  4. $24\pi$
  5. $36\pi$
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E Correct answer
Explanation

The equation y = sqrt(6x - x^2) is y^2 = 6x - x^2, or (x-3)^2 + y^2 = 9. This is a semicircle with radius 3 centered at (3, 0). Rotating this about the x-axis creates a sphere of radius 3. Volume = (4/3) * pi * r^3 = (4/3) * pi * 27 = 36 * pi.

AI explanation

The equation y = sqrt(6x - x^2) can be rewritten as y = sqrt(9 - (x - 3)^2), showing the region is a semicircle with a radius of 3 centered at (3, 0). Rotating this semicircle 360 degrees about the x-axis generates a full sphere of radius 3. The volume of a sphere is given by the formula V = 4/3*pi*r^3. Substituting r = 3 gives V = 4/3*pi*(27) = 36*pi.

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