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05 Aug 2019, 01:19
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[GMAT math practice question]
As the figure shows, a ball and cone are inscribed by a cylinder. What is the ratio of the volumes of the sphere to the cone to the cylinder?
8.5 PS.png [ 57.4 KiB | Viewed 1790 times ]
\(A. 1:1:2\)
\(B. 1.5:1:3\)
\(C. 1.5:1.5:2\)
\(D. 2:1:3\)
\(E. 3:2:4\)
As the figure shows, a ball and cone are inscribed by a cylinder. What is the ratio of the volumes of the sphere to the cone to the cylinder?
Attachment:
8.5 PS.png [ 57.4 KiB | Viewed 1790 times ]
\(A. 1:1:2\)
\(B. 1.5:1:3\)
\(C. 1.5:1.5:2\)
\(D. 2:1:3\)
\(E. 3:2:4\)
05 Aug 2019, 04:38
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This question is pretty straight forward if you know the formulas and realize that the diameter of the sphere is equal to the height of the cone and cylinder.
Vol of sphere : 4/3*(pi)*r^3
Vol of cone : 1/3*pi*r^2*h
Vol of cylinder : pi*r^2*h
pi and r^2 pretty much cancel right away. 2r = h (as mentioned above)
therefore, we get - 2:1:3
Vol of sphere : 4/3*(pi)*r^3
Vol of cone : 1/3*pi*r^2*h
Vol of cylinder : pi*r^2*h
pi and r^2 pretty much cancel right away. 2r = h (as mentioned above)
therefore, we get - 2:1:3
07 Aug 2019, 03:15
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=>
From the figure, the radii of the sphere and the base circle are \((\frac{9}{2})\).
The volume of the sphere is \((\frac{4}{3}) π(\frac{9}{2})^3 = π(\frac{243}{2}).\)
The volume of the cone is \((\frac{1}{3}) π(\frac{9}{2})^2(9) = π(\frac{243}{4}).\)
The volume of the cylinder is \(π(\frac{9}{2})^2(9) = π(\frac{729}{4})\)
Thus, the volumes are in the ratio \(π(\frac{243}{2}) : π(\frac{243}{4}) : π(\frac{729}{4}) = 2 : 1 : 3.\)
Therefore, the answer is D.
Answer: D
From the figure, the radii of the sphere and the base circle are \((\frac{9}{2})\).
The volume of the sphere is \((\frac{4}{3}) π(\frac{9}{2})^3 = π(\frac{243}{2}).\)
The volume of the cone is \((\frac{1}{3}) π(\frac{9}{2})^2(9) = π(\frac{243}{4}).\)
The volume of the cylinder is \(π(\frac{9}{2})^2(9) = π(\frac{729}{4})\)
Thus, the volumes are in the ratio \(π(\frac{243}{2}) : π(\frac{243}{4}) : π(\frac{729}{4}) = 2 : 1 : 3.\)
Therefore, the answer is D.
Answer: D
