Question

1. The barrel of a fountain pen, cylindrical in shape, is 7cm long and 5mm in diameter. A full barrel of ink in the pen will be used upon writing 3300 words on an average. How many words can be written in a bottle of ink containing one fifth of a litre?
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2. Water flow at the rate of 10 meters per minute through a cylindrical pipe having the diameter as 5mm. How much time will it take to fill a conical vessel whose diameter of base is 40cm and depth24cm?
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Answer 1

=》 Volume of a barrel = 22/7×0.25×0.25×7 = 1.375 cm^3

= 》Volume of ink in the bottle = 1/5 litre = 1000/5 = 200 cm ^3

=》Therefore, total number of barrels that can be filled from the given volume of ink = 200/1.375

=》So, required number of words = 200/1.375×330 = 48000

☆ Hope it helps ☆

Answer 2

${\underline{\bf{\underline{Solution\:1}}}}$

$\bf{Given,}$

• Length of the barrel of the fountain pen = 7cm
• Diameter of the barrel of the fountain pen = 5mm = 0.5cm (as 1mm = 10cm)

$\bf{To\:find,}$

• Total number of words that can be written in a bottle of ink containing one fifth of a litre.

$\bf{Solution,}$

If the diameter of the barrel is 0.5cm then, the radius would be $\sf{\frac{0.5}{2}cm=0.25cm}$  because as we know Radius = $\sf{\frac{Diameter}{2}}$

Now, the volume of the cylindrical barren = $\sf{\pi r^2h}$

Putting the respected diameter and radius as given in the question -

Note => Putting, π = $\sf{\frac{22}{7}}$

    $\:\:\:\:\implies\sf{the\:volume\: of\: the\: cylindrical\: barren=\frac{22}{7}\times 0.25^2\times 7 cm^3}$

    $\:\:\:\:\implies\bf{the\:volume\: of\: the\: cylindrical\: barren= 1.375cm^3}$

Now calculating, the volume of ink in the bottle =  $\sf{{\frac{1}{5}}^{th}\:of\:a\:litre}$

   $\:\:\:\:\implies\sf{the\:volume\: of\: ink\:in\:the\: bottle=\frac{1}{5} \times1000cm^3 }$  (as, 1 litre = 1000cm³)

   $\:\:\:\:\implies\bf{the\:volume\: of\: ink\:in\:the\: bottle= 200 cm^3}$

Now,

If 1.375 cm³ ink is used for writing number of words = 3300 words

then, 1 cm³ ink is used for writing number of words = $\bf{\frac{3300}{1.375}\:}$ words

${\underline{\therefore\bf{200 cm^3\: ink\: is \:used\: for\: writing\: number\: of \:words=\frac{3300}{1.375} \times 200=480000}}}$

${\underline{\bf{\underline{Solution\:2}}}}$

$\bf{Given,}$

• Volume of the water that flows out in one minute.
• Volume of the cylinder of diameter 5 mm and length 10 metre.
• Volume of the cylinder of radius $\sf{\frac{5}{2}mm=\frac{1}{4}cm}$.
• Volume of the cylinder of length 1000cm = $\sf{\frac{22}{7}\times\frac{1}{4}\times\frac{1}{4}\times1000cm^3}$

$\bf{To\:find,}$

• Total time will it take to fill a conical vessel.

$\bf{Solution,}$

Volume of a conical vessel of base radius 20 cm and depth 24 cm = $\sf{\frac{1}{3}\times\frac{22}{7}\times(20)^2\times24\:cm^3}$

Let, the conical vessel is filled in x minutes.

$\therefore$ Volume of the water that flows out in x minutes = Volume of the conical vessel

     $\:\:\:{\implies{\sf{\frac{22}{7}\times\frac{1}{4}\times\frac{1}{4}\times1000\times x =\frac{1}{3}\times\frac{22}{7}\times20^2\times24}}}$

     $\:\:\:\implies{\sf{x=\frac{1}{3}\times\frac{400\times24\times4\times4}{1000}=\frac{512}{10} minutes}}$

     $\:\:\:\implies{\bf{x=51\:minutes\:12\:seconds}}$

$\therefore{\underline{\bf{Total\: time\: will \:it\: take\: to\: fill\: a \:conical\: vessel= 51\:minutes\:12\:seconds}}}$

${\underline{\bf{\underline{Formulae\:Applied}}}-}$

$\bf{\bullet}$ Volume of cylinder = $\bf{h\pir^2}$

where,

h = height of the cylinder

r = radius of the base

$\bf{\bullet}$ Volume of circular cone =  $\bf{\frac{1}{3}\pi r^2h}$

where,

h = height of the cylinder

r = radius of the base