Solve Weird Depth Problem: Conical Tank Water Flow Rate

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In summary, the conversation is about a related rates problem involving a conical tank with water flowing in at a rate of 10 cubic feet per minute. The task is to find the rate of change of the depth of the water when it is 8 feet deep. The formula for the volume of a cone in terms of its radius and height is needed to solve the problem, with the given tank having a height of 12 feet and a radius of 5 feet. The conversation concludes with the person being reassured that they should be able to solve the problem since they just started learning about related rates.
  • #1
calcatmillbrook
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I've looked everywhere to try to solve this problem and I can't find anything. It is:

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rateof 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

Thanks for the help.
 
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  • #2
this is a pretty easy related rates problem
 
  • #3
Why would you consider that weird? In general, if you can find a formula for volume as a function of height, V(h), then, by the chain rule, dV/dt= (dV/dh) dh/dt. You are given dV/dt and asked to find dh/dt. You need to be able to calculate dV/dh. Do you know (or can you look up) the formula for volume of a cone in terms of radius and height? Here you are told that the tank itself has height 12 feet and radius 5 feet. Do you see that any "cone of water" contained by that tank will have h/r= 12/5?
 
  • #4
sorry, i just started rates on friday
 
  • #5
Good! Then you should know exactly how to do this!
 

What is a conical tank?

A conical tank is a tank with a circular base and a conical or sloping sides that converge to a point at the top. It is typically used for storing liquids or powders and is often used in industrial processes.

What is the depth problem in a conical tank?

The depth problem in a conical tank refers to the difficulty in accurately measuring the volume of liquid in the tank due to the varying depth caused by the sloping sides. This can make it challenging to determine the flow rate of liquid entering or leaving the tank.

How do you solve the depth problem in a conical tank?

To solve the depth problem in a conical tank, you can use the formula V = (πr2h)/3, where V is the volume of the tank, r is the radius of the base, and h is the height or depth of the liquid. This formula takes into account the varying depth caused by the sloping sides of the tank.

What factors can affect the flow rate in a conical tank?

The flow rate in a conical tank can be affected by several factors, including the size and shape of the tank, the viscosity of the liquid, the pressure and temperature of the liquid, and the size and placement of the inlet and outlet pipes. Other factors such as turbulence and surface tension can also impact the flow rate.

How can you calculate the flow rate in a conical tank?

To calculate the flow rate in a conical tank, you can use the formula Q = A * v, where Q is the flow rate, A is the cross-sectional area of the tank at a specific point, and v is the average velocity of the liquid at that point. This formula can be modified to account for any changes in depth caused by the sloping sides of the tank.

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