Related Rates in a water tank

In summary, a water tank in the shape of an inverted right-circular cone with a top radius of 15 meters and depth of 12 meters is being filled with water at a rate of 2 cubic meters per minute. The problem is to determine the rate at which the depth of water is increasing when it reaches 8 meters. Using the formula for volume of a cone and the given rate of change for height, we can set up an equation to relate the rate of change of volume to the rate of change of height and radius. By finding a way to express the radius in terms of height, we can solve for the rate of change of height at the given depth of water. Further steps are needed to find the final solution.
  • #1

Homework Statement


A water tank has the shape of an inverted right-circular cone, with radius at the top 15 meters and depth 12 meters. Water is flowing into the tank at rate of 2 cublic meters per minute. How fast is the depth of water in the tank increasing at the instant when the depth is 8 meters


Homework Equations


V=(1/3)(pi)(r^2)(h)

The Attempt at a Solution


dv/dt=2 meter^3/min
dv/dt=(1/3)(pi)(2rh(dr/dt)+(r^2)(dh/dt))
but that has 2 unknow varibles in it.
 
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  • #2
You need to relate the radius in terms of the height of the cone since both are changing with respect to time and because you only know the rate of change of the height. Once you do this, you will be able to differentiate it since you know the rate of change of the height. Try to find a way to relate the radius in terms of height. With this information can you figure out how to get rid of r and put it in terms of h?
 
  • #3
Yeah thanks that helps a lot
 

1. What is meant by "related rates" in a water tank?

Related rates in a water tank refers to the concept of how the rate of change of one variable (such as the water level in the tank) affects the rate of change of another related variable (such as the volume of water in the tank).

2. How do you calculate related rates in a water tank?

To calculate related rates in a water tank, you must first identify the variables involved and the relationship between them. Then, you can use the chain rule from calculus to find the derivative of the related variables and set up an equation to solve for the desired rate of change.

3. What are some common applications of related rates in a water tank?

Related rates in a water tank can be applied in various situations, such as calculating the flow rate of water into or out of the tank, determining the rate of change of the water level, or finding the rate at which the volume of water is changing.

4. Are there any limitations to using related rates in a water tank?

Yes, there are limitations to using related rates in a water tank. These may include assuming a constant rate of change, neglecting factors such as evaporation or leaks, and assuming a perfectly cylindrical or rectangular tank shape.

5. Can related rates in a water tank be applied to other fluid systems?

Yes, the concept of related rates can be applied to other fluid systems, such as pipes, tanks, and pumps. However, the specific equations and variables involved may differ, and certain assumptions may need to be made depending on the system.

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