A conical tank is of height 12 m and surface diameter 8m. Water is pumped into the tank at the rate of 50 m^3/(min). How fast is the water level increasing when the depth of the water is 6 m?

Tyra 2020-11-16 Answered
A conical tank is of height 12 m and surface diameter 8m. Water is pumped into the tank at the rate of 50m3min. How fast is the water level increasing when the depth of the water is 6 m?
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Expert Answer

Arnold Odonnell
Answered 2020-11-17 Author has 109 answers
Step 1
27 (a)Given- height(H)=12 m, diameter(D)=8m and dVdt=50m3min.
To find- the level of the water increasing when the depth of the water 6m.
Formula Used- the volume of the conical tank (V)=13πr2h, when , r= radius and h=height.
Step 2
Explanation- The volume of the tank (V)=13πr2h ⋯⋯(1)
As we have to find the level of the water increasing when the depth of the water 6m, so we have to eliminate the radius (r) in terms of h, which can be written as follws,
HR=hr (using the triangle similarity)
Further, we can write as,
r=RH·h
Now, substituting the value of r in equation (1), we get,
V=13π(RHh)2h
V=13πR2H2h3
Step 3
Now, diffferentiating the above expression, w.r.t. h, we get,
dVdt=13πR2H23h2dhdt
50=π(412)2(6)2dhdt
Further, we can write as,
dhdt=504π
=3.9784mmin
So, the level of the water increasing when the depth of the water 6m at 3.9784mmin.
Result: Hence, the level of the water increasing when the depth of the water 6m at 3.9784mmin.
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