Question : Assume that a drop of water is spherical and that its diameter is one-tenth of a cm. A conical glass has a height equal to the diameter of its rim. If 32,000 drops of water fill the glass completely, then the height of the glass (in cm) is:
Option 1: 1
Option 2: 2
Option 3: 3
Option 4: 4
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Correct Answer: 4
Solution :
Height of the glass $=h$ cm
Radius of the glass $=\frac{h}{2}$
Volume of the glass $=\frac{1}{3}\pi r^2 h=\frac{1}{3}\pi (\frac{h}{2})^2 h$
Radius of the spherical drop $=\frac{1}{2\times10}=\frac{1}{20}$ cm
Volume of a spherical drop $=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (\frac{1}{20})^3$
Volume of 32,000 spherical drops $=32000\times\frac{4}{3}\pi (\frac{1}{20})^3$
Now, the volume of the glass = volume of 32000 drops
So, $\frac{1}{3}\pi r^2 h=32000\times\frac{4}{3}\pi (\frac{1}{20})^3$
⇒ $\frac{h^3}{4}=32000\times4\times\frac{1}{8000}$ [$\because r = \frac{h}{2}$]
⇒ $h^3 = 64$
$\therefore h = 4\ \text{cm}$
Hence, the correct answer is 4.
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