Problem 23 · 2012 Math Kangaroo
Stretch
Geometry & Measurement
area
Let a > b. If the ellipse shown rotates about the x-axis an ellipsoid Ex with volume Vol(Ex) is obtained. If it rotates about the y-axis an ellipsoid Ey with volume Vol(Ey) is obtained. Which of the following statements is true?
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Answer: C — Ex ≠ Ey and Vol(Ex) > Vol(Ey)
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Hint 1 of 2
Read the figure: the ellipse is tall, with the larger semi-axis \(a\) along the \(y\)-axis and the smaller \(b\) along the \(x\)-axis.
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Hint 2 of 2
When you spin a shape about an axis, the volume grows with the square of the radius (the semi-axis perpendicular to the spin axis).
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Approach: compare the two solids of revolution
- From the figure the ellipse has semi-axis \(a\) up the \(y\)-axis and \(b\) along the \(x\)-axis, with \(a > b\).
- Spinning about the \(x\)-axis sweeps radius \(a\), giving a solid with semi-axes \((b,a,a)\) and volume \(\tfrac{4}{3}\pi a^2 b\); spinning about the \(y\)-axis sweeps radius \(b\), giving semi-axes \((b,b,a)\) and volume \(\tfrac{4}{3}\pi a b^2\). The two solids clearly differ.
- Since \(a > b\), \(\tfrac{4}{3}\pi a^2 b > \tfrac{4}{3}\pi a b^2\), so \(\mathrm{Vol}(E_x) > \mathrm{Vol}(E_y)\).
- Thus \(E_x \ne E_y\) and \(\mathrm{Vol}(E_x) > \mathrm{Vol}(E_y)\), choice C.
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