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CBSE Class 12-science Answered

Look at the conical funnel whose vertical angle is 90o. Water is dripping out from it at uniformly at the rate of 8 cm3/sec through a tiny hole at the vertex.  Find the rate at which the slant height of the water is decreasing when the slant height of the water is 4cm. Is leaking of water leads to wastage of water? Do you think we can afford to waste our natural resources?
Asked by Topperlearning User | 05 Aug, 2014, 01:35: PM
Expert Answer

Let V be the volume of the Water at time 't'.

 

Then,straight V equals 1 third πr squared straight h where 'r' and 'h' are the radius and height respectively.  From the figure it is clear 

r equals l space sin space 45
h equals l space cos space 45

Therefore the volume is

 V equals 1 third straight pi left parenthesis straight l space sin space 45 right parenthesis squared straight l space cos space 45
equals 1 third straight pi open parentheses fraction numerator straight l over denominator square root of 2 end fraction close parentheses squared fraction numerator straight l over denominator square root of 2 end fraction equals fraction numerator πl cubed over denominator 6 square root of 2 end fraction

                

Differentiating with respect to t we get

dv over dt equals fraction numerator 3 straight pi over denominator 6 square root of 2 end fraction straight l squared. dl over dt

 Now we are given dv over dt equals minus 8 space cm cubed divided by sec

Therefore, dl over dt equals minus fraction numerator 16 square root of 2 over denominator πl squared end fraction

When the slant height is 4cm

dl over dt equals fraction numerator square root of 2 over denominator straight pi end fraction cm divided by sec

Slant height is decreasing at the rate of

fraction numerator square root of 2 over denominator straight pi end fraction cm divided by sec

Yes, leaking of water leads to wastage of water. We must do everything possible to save our natural resources.

Answered by | 05 Aug, 2014, 03:35: PM
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