Waste is being pumped into a closed tank buried in the ground that has the shape

Question

Waste is being pumped into a closed tank buried in the ground that has the shape

of an inverted circular cone with a maximum radius of 2 m and height 4 m. The

city sanitation department wants to know the rate at which the waste level is rising

in the tank but they are they are unable to crawl inside and take any measurements.

They do know that waste is pumped in at a rate of 2 m3 / min. At what rate is the

waste level rising when the depth of the waste is 3 m. Recall that the equation for

volume of a cone

Explanation / Answer

V=pi*R^2*H/3

As R=2 and H=4

So

semi-vertical angle=tan^-1(2/4)=26.565 degrees

Let at a distance h radius be r

So

r=htan(26.565)=h/2

and

dr/dt=0.5dh/dt

So

dV/dt=2pi*r*h/3*dr/dt +pi*r^2*(dh/dt)/3=[(1/6)*pi*h^2+(1/12)pi*h^2]*dh/dt

So

we have at h=3 and dV/dt=2

So

2=pi*(9/4)*dh/dt

So

dh/dt=8/(9pi)=0.283 m/min

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