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eswarchethu135
GMAT 2: 640 Q49 V27
Posts: 276
03 Nov 2018, 09:37
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
40% (02:13) correct
60%
(02:18)
wrong
based on 83
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Attachment:
image.jpg [ 18.5 KiB | Viewed 7066 times ]
In the figure above, when a conical bottle is rested on its flat base, the water in the bottle is 8 cm from its vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the approximate height of the bottle?
A. 5cm
B. 8cm
C. 10cm
D. 12cm
E. 15cm
eswarchethu135
GMAT 2: 640 Q49 V27
Posts: 276
04 Nov 2018, 03:08
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Attachment:
image 1.jpg [ 32.27 KiB | Viewed 6951 times ]
Consider the image shown above.
Let the height of the cone be "h" and "R" be the radius of the cone. When the bottle is rested on its flat base, the volume of water is given by
Volume of the cone - volume of empty space above the water level
\(\frac{\pi}{3} R^2 h\) - \(\frac{\pi}{3} r_1^2 (8)\)
Now by similar triangle concept:
\(\frac{r_1}{8} = \frac{R}{h}\)
\(r_1 = \frac{8R}{h}\)
Sub \(r_1\) value in the above equation, we get
volume of water = \(\frac{\pi}{3} R^2 h - \frac{\pi}{3} (\frac{8R}{h})^2 (8)\)
\(\frac{\pi}{3} R^2 (h - \frac{512}{h^2})\) ......................................... (1)
Now consider the cone is tilted upside down. Volume of water is
\(\frac{\pi}{3} (r_2)^2 (h-2)\)
Again by similar triangle concept:
\(\frac{r_2}{h-2} = \frac{R}{h}\)
\(r_2 = \frac{(h-2) R}{h}\)
Substituting \(r_2\) in the above equation of volume of water when cone is tilted,
volume of water = \(\frac{\pi}{3} (\frac{(h-2) R}{h})^2 (h-2)\)
\(\frac{\pi}{3} R^2 \frac{(h-2)^3}{h^2}\) ..................................... (2)
Since volume of water in both the cases is equal, (1) = (2)
\(\frac{\pi}{3} R^2 (h - \frac{512}{h^2})\) = \(\frac{\pi}{3} R^2 \frac{(h-2)^3}{h^2}\)
\(h - \frac{512}{h^2} = \frac{(h-2)^3}{h^2}\)
\(h^3 - 512 = (h-2)^3\)
\(h^3 - 512 = h^3 - 6h^2 + 12h - 8\)
\(6h^2 - 12h - 504 = 0\)
\(h^2 - 2h - 84 = 0\)
Since \(h > 0 , h \approx {10 cm}\)
OPTION : C
05 Nov 2018, 13:03
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Hi,
Just thinking out loud (and this may sound like a stupid question). Would I be wrong if I just went straight ahead to add the height of the spaces 8cm + 2cm ?
cc: Bunuel, GMATNinja
PS: I've been struggling with the compulsion to rush into doing extreme math on my test prep and I am try to break away from that habit.
Just thinking out loud (and this may sound like a stupid question). Would I be wrong if I just went straight ahead to add the height of the spaces 8cm + 2cm ?
cc: Bunuel, GMATNinja
PS: I've been struggling with the compulsion to rush into doing extreme math on my test prep and I am try to break away from that habit.
