Respuesta :
The inflow rate of water is given by the function of the volume of the
tank derived from the shape of the tank.
Response:
- The rate water is being pumped into the tank is approximately 286,253 cm³/min
Which method can be used to find the rate at which water is pumped into the tank?
The given parameter are;
Shape of the tank = Conical
Rate at which water is leaking out of the tank = 7,000 cm³/min
Height of the tank = 6 m
Diameter of the tank = 4 m
Solution:
We have;
[tex]\dfrac{D}{h} = \mathbf{\dfrac{2 \cdot r}{h}}[/tex]
[tex]r = \mathbf{\dfrac{D}{2 \cdot h}}[/tex]
Which gives;
[tex]\dfrac{r}{h} = \dfrac{4}{6 \times 2} = \dfrac{1}{3}[/tex]
[tex]r= \mathbf{\dfrac{h}{3}}[/tex]
[tex]V = \dfrac{1}{3} \cdot \pi \cdot r^2 \cdot h = \dfrac{1}{3} \cdot \pi \cdot \left(\dfrac{h}{3} \right)^2 \cdot h = \mathbf{ \pi \cdot \dfrac{h}{27} ^3}[/tex]
Which gives;
[tex]\dfrac{dV}{dh} = \mathbf{\pi \cdot \dfrac{h}{9} ^2}[/tex]
By using chain rule of differentiation, we have;
[tex]\dfrac{dV}{dt} = \mathbf{\dfrac{dV}{dh} \times \dfrac{dh}{dt}}[/tex]
Which gives;
[tex]\dfrac{dV}{dt} = \mathbf{\pi \cdot \dfrac{h}{9} ^2\times 20}[/tex]
[tex]\dfrac{dV}{dt} = \pi \cdot \dfrac{200}{9} ^2\times 20 \approx 279,252.68[/tex]
[tex]\dfrac{dV}{dt} = \mathbf{Inflow \ rate - Outflow \ rate}[/tex]
Therefore;
[tex]Inflow \ rate = \mathbf{\dfrac{dV}{dt} + Outflow \ rate}[/tex]
Which gives;
Inflow rate ≈ 279,252.68 + 7000 = 286,252.68 ≈ 286,253
- The inflow rate = The rate water is being pumped into the tank ≈ 286,253 cm³/min
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