NIMCET 2009 Question Paper with solutions

Question : 11 of 120

Marks: +1, -0

Water runs into a conical tank of radius 5 ft and height 10 ft , at a constant rate of

2 \ft ^ 3

/ min . How fast is the water level rising when the water is 6 ft deep?

Solution:

Volume of tank,

V=\;{1}/{3} π i^{2} h

....(i)
Also:

\;{h}/{r}=\;{10}/{5}=2 ⇒ h=2 r

.....(ii)
Now,

\;{d V}/{d t}=\;{π}/{3}[r^{2} \;{d h}/{d t}+2 r h \;{d r}/{d r}]

....(iii)
Also,

\;{d h}/{d t}=2 \;{d r}/{d t}

Then, Eq. (iii) becomes

⇒ \;\; \;{d V}/{d t}=\;{π}/{3}[\;{d h}/{d t}(r^{2}+r h)]

⇒ \;\; 2=\;{π}/{3}[\;{d h}/{d t}(9+18)]

( ∵ h=6, x=3

and

\;{d V}/{d t}=2)

\;{d h}/{d t}=\;{2}/{9 π} {\ft} {/ }{\min}

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