Conical Tank Water Leak Rate Calculation

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In summary, the water level is rising at a rate of 20cm/min, but the rate at which water is being pumped into the tank is only 10,000 cm^3 / min.
  • #1
Feodalherren
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Homework Statement



Water is leaking out of an inverted conical tank at a rate of 10,000 cm^3 / min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find the rate at which water is being pumped into the tank.

Homework Equations





The Attempt at a Solution



This is how I started:

I want dV/dt when h=200 and dh/dt = 20.

I used similar triangles to get the radius of the smaller cone to be 1/√8

The volume of a cone is:
V=(1/3)∏hr^2

Last step was simply to differentiate the volume formula with the radius. Somewhere something went wrong, it just feels wrong to me... Help? :)
 
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  • #2
Feodalherren said:

Homework Statement



Water is leaking out of an inverted conical tank at a rate of 10,000 cm^3 / min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find the rate at which water is being pumped into the tank.

Homework Equations





The Attempt at a Solution



This is how I started:

I want dV/dt when h=200 and dh/dt = 20.

I used similar triangles to get the radius of the smaller cone to be 1/√8
This is where you went wrong. The radius of the smaller cone is changing all the time. Use similar triangles to get a relationship between the radius and height of the smaller cone. Then you can write the volume as a function of either h or r alone.
Feodalherren said:
The volume of a cone is:
V=(1/3)∏hr^2

Last step was simply to differentiate the volume formula with the radius. Somewhere something went wrong, it just feels wrong to me... Help? :)
 
  • #3
Thank you I got it now! :)
 

FAQ: Conical Tank Water Leak Rate Calculation

Why is water leaking out of a cone?

Water leaks out of a cone because of the force of gravity. The water inside the cone is pulled downwards towards the bottom, causing it to leak out of the opening at the bottom of the cone.

How does the shape of the cone affect water leakage?

The shape of the cone plays a major role in water leakage. The narrow tip of the cone concentrates the force of gravity on a smaller area, resulting in a faster flow of water. The wider opening at the bottom of the cone allows for more water to flow out at once.

Can the rate of water leakage be controlled?

Yes, the rate of water leakage can be controlled by adjusting the size of the opening at the bottom of the cone. A smaller opening will slow down the flow of water, while a larger opening will increase it.

What other factors can affect water leakage from a cone?

The viscosity of the water and the material of the cone can also affect water leakage. Thicker, more viscous liquids will take longer to leak out of the cone. Additionally, if the cone is made of a porous material, such as paper, it may absorb some of the water, resulting in slower leakage.

Is there a limit to how much water can leak out of a cone?

Yes, there is a limit to how much water can leak out of a cone. Once the water level inside the cone reaches the opening at the bottom, the force of gravity will no longer be able to pull the water downwards and it will stop leaking. Additionally, if the cone is not completely sealed, some water may also leak out through small gaps or cracks in the cone.

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