How do you find the rate at which water is pumped into an inverted conical tank that has a height of 6m and a diameter of 4m if water is leaking out at the rate of #10,000(cm)^3/min# and the water level is rising #20 (cm)/min#?

Question

Emma Aldridge · Accepted Answer

This question has already been answered although you seem to be missing the height of the water in the cone at the time the water level is rising #20 (cm)/(min)#.
Assuming this question came from the same source, the specified height of water was #2 m# or #200 cm#. The cone has a radius of 2 m (half the diameter) and a height of 6 m
for a ratio of #(radius)/(height) = 1/3# This ratio is constant for volumes of water contained in the cone, Therefore the volume of the cone (or water in the cone), normally written as
#V(r,h) = (pi r^2h)/3# can be re-written as
#V(h) = ( pi (h/3)^2 h)/3#
#= (pi h^3)/(27)# and therefore
#(d V(h))/(dh) = pi/9 h^2# #(cm^3)/(cm)# We are told 
#(d  h)/(dt) = 20 (cm)/(min)# The increase in volume contained in the cone is given by
#(d V)/(dh) xx (d h)/(dt)# at water level height of #200 cm# #= pi/9 (200 cm)^2 xx 20 (cm)/(min)#
#= 2,792,527 (cm^3)/(min)# (approx. assuming I haven't slipped up somewhere) The inflow of water must be the total of 
the outflow (leakage)
plus
the amount needed to raise the water level:
#10,000 ((cm)^3)/(min)#
#+ 2,792,527 ((cm)^3)/(min)# #= 2,802,527 ((cm)^3/min)#

Axel Alvarez · Answer

undefined To find the rate at which water is pumped into the inverted conical tank, use the formula:

Rate of pumping = Rate of rise in water level - Rate of leakage

First, convert all measurements to the same units. Since the rate of leakage is given in cubic centimeters per minute (cm³/min) and the rate of rise in water level is given in centimeters per minute (cm/min), they are already in the same unit.

The volume of a cone is given by the formula: Volume = (1/3) * π * r² * h

Given that the tank's height (h) is 6m and diameter (d) is 4m, the radius (r) can be found using the formula: r = d/2

Substitute the values into the volume formula to find the volume of the tank.

Now, differentiate the volume formula with respect to time (t) to find the rate of change of volume with respect to time (dV/dt).

Finally, substitute the given rates (rate of leakage and rate of rise in water level) into the formula: Rate of pumping = Rate of rise in water level - Rate of leakage

Solve for the rate of pumping to get the answer.