Problem 1 :
The barrel of a fountain-pen cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used for writing 330 words on an average. How many words can be written using a bottle of ink containing one fifth of a liter?
Solution :
To find the number of words can be written using a bottle of ink, we have to find the quantity of ink in the barrel.
Volume of barrel in the cylindrical shape = πr^{2}h
height of shape = 7 cm
radius of the shape = (5/2) mm
10 mm = 1 cm
(5/2)/10 = (1/4) cm
Volume of barrel in the cylindrical shape :
= (22/7) ⋅ (1/4) (1/4) 7
= (11/8) cm^{3}
1000 cm^{3} = 1 liter
= 11/8000 liter
Quantity of ink in 1 liter = 8000/11
Quantity of ink in the bottel = (1/5) of liter of ink in the barrel
= (1/5) (8000/11)
Number of word written using 1 bottel of ink = 330
Required number of words = (1/5) (8000/11) ⋅ (1/4)
= 48000 words
Problem 2 :
A hemi-spherical tank of radius 1.75 m is full of water. It is connected with a pipe which empties the tank at the rate of 7 liters per second. How much time will it take to empty the tank completely?
Solution :
To solve this problem, first let us find quantity of water in the tank in liters.
Volume of water in the hemispherical tank = (2/3)πr^{3}
= (2/3) ⋅ (22/7) ⋅ (1.75)^{3}
= (2/3) ⋅ (22/7) (1.75) (1.75) (1.75)
= 235.81/21
= 11.22 m^{3}
1 m^{3} = 1000 liter
= 11.22 (1000)
= 11220 liters
Quantity of water empties = 7 liter per second
= 7 (60)
= 420 liter per minute
Time taken to empty the tank = 11220/420
= 26.71
= 27 minutes (approximately)
Problem 3 :
Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r units.
Solution :
Volume of cone craved out from the hemisphere
= (1/3) r^{2}h
= (1/3) r^{2}r
= (1/3) r^{3}
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