Related rates, conic tank

In summary, the question asks for the exact value of the rate of change of the water level in a cone-shaped tank when the water height is 3/4 the height of the cone. To solve this, we use the formula for volume of a cone and trigonometry to find the radius at the given water level. Then, we take the derivative of the volume with respect to time and plug in the given rate of water flow to solve for the rate of change of the water level. The final answer is (64)/(9*pi*h^2), where h represents the height of the cone.
  • #1
shanshan
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Homework Statement


Water si emptying out of a tank at a rate of 4 cubic metres/min. The tank is shaped like an inverted circular cone with a diameter of 8 m at the top. The sides of the cone make a 30 degree with the vertical. Determine the exact value of the rate of change of the water level when the height of the water is 3/4 the height of the cone.


Homework Equations


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


The Attempt at a Solution


v = (1/3) (pi) (3/4(h))^2 (h)
=(3/16)(pi)(h^3)
v'=(9/16)(pi)(h^2)(h')

i don't know where to go from here, as i could use trig to get the height of the tank, but i don't have the height at the time specified. help?
 
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  • #2


Thank you for your question. To find the exact value of the rate of change of the water level, we need to use the formula for the volume of a cone, which you have correctly identified as v=(1/3)(pi)(r^2)(h). However, we need to make sure that all the units are consistent, so we will use metres for length and cubic metres for volume.

First, let's find the radius of the cone at the water level when it is 3/4 the height of the cone. We can use trigonometry to find this value. Since the sides of the cone make a 30 degree angle with the vertical, we can use the sine function to find the ratio of the opposite side (radius at the water level) to the hypotenuse (height of the cone).

sin(30) = radius at water level / height of cone
0.5 = radius at water level / 8 m
radius at water level = 4 m

Now, we can plug this value into our original formula for volume to find the volume of water in the cone at this height.

v = (1/3)(pi)(4^2)(3/4h)
v = (4/3)(pi)(3/4h)^3
v = (1/3)(pi)(3/4h)^3

Next, we can take the derivative of this volume with respect to time to find the rate of change of the volume of water in the cone.

v' = (1/3)(pi)(3/4h)^2 (3h'/4)
v' = (9/16)(pi)(h^2)(h')

Finally, we can plug in the given rate of water flow of 4 cubic metres per minute to solve for the rate of change of the water level.

4 = (9/16)(pi)(h^2)(h')
h' = (4*16)/(9*pi*h^2)
h' = (64)/(9*pi*h^2)

We don't have a specific height given, so we will leave our answer in terms of h. However, if we wanted to find the rate of change at a specific height, we could plug that value in for h. I hope this helps! Let me know if you have any further questions.



Scientist
 

1. What is a conic tank and how does it relate to related rates?

A conic tank is a type of storage tank that has a conical shape, with a circular or elliptical base and sloping sides. In related rates problems, the volume of a conic tank is often used as a variable to represent the rate of change of a certain quantity, such as the height or diameter of the tank.

2. How do you set up a related rates problem involving a conic tank?

To set up a related rates problem involving a conic tank, you first need to identify the variables and their rates of change. Then, you can use the formula for the volume of a conic tank (V = 1/3 * π * r^2 * h) to create an equation that relates the variables. Finally, you can use implicit differentiation to find the rate of change of the desired variable.

3. Can the shape of the conic tank affect the related rates problem?

Yes, the shape of the conic tank can affect the related rates problem. For example, if the tank has a elliptical base rather than a circular one, the formula for the volume will be different and the related rates problem will need to be set up accordingly.

4. What are some real-life applications of related rates problems involving conic tanks?

Related rates problems involving conic tanks can be found in many real-life scenarios, such as calculating the rate at which water is draining from a conic tank, determining the rate at which a chemical is being added to a conic tank, or finding the rate at which the dimensions of a conic tank are changing due to expansion or contraction.

5. Are there any limitations to using related rates with conic tanks?

One limitation to using related rates with conic tanks is that it assumes the tank is a perfect shape with no irregularities or imperfections. In real-life scenarios, this may not always be the case, so the results may not be entirely accurate. Additionally, related rates problems involving conic tanks can become quite complex and require advanced mathematical skills to solve.

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