Differential Calculus - Related Rates

In summary: Thank you, again!In summary, the problem involves finding a formula to relate the tax of variation of liquid height with the height. By using the given information, the formula is determined to be dV/dt = pi*3*y^2/48 dy/dt = -32/pi*y^2.
  • #1
ruiwp13
40
0

Homework Statement



A liquid is being filtrated by a filter with a cone form. The filtring tax is 2cm^3/min. The cone has 16cm height, 4 cm radius. The volume V is given by pi*r^2*y/2 where y is the height, r the radius. Discover a formula that relates the tax of variation of the liquid height with the height (y I guess).

Homework Equations



dV/dt = V*dh/dt
V=pi*r^2*y/3

The Attempt at a Solution



the volume is = pi*r^2*y/3

so Volume = pi.(4^2)*y/3

Volume=16*pi*y/3

dv/dt = 16*pi/3*dh/dt

-2 = 16*pi/3 dh/dt

dh/dt = -6/16*pi

But this solution is incorrect :/ Can anyone tell me where I missed?

Thanks in advance
 
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  • #2
welcome to pf!

hi ruiwp13! welcome to pf! :smile:

(try using the X2 button just above the Reply box :wink:)
ruiwp13 said:
the volume is = pi*r^2*y/3

so Volume = pi.(4^2)*y/3

nooo :redface: … r is a function of y, not fixed

so the volume is a multiple of y3 :wink:
 
  • #3


tiny-tim said:
hi ruiwp13! welcome to pf! :smile:

(try using the X2 button just above the Reply box :wink:)


nooo :redface: … r is a function of y, not fixed

so the volume is a multiple of y3 :wink:

didn't follow you there :/ what do you mean?
 
  • #4
the volume is a function of the radius (of the liquid surface) at time t, and the height at time t

the radius at time t is not 4, it depends the height :wink:
 
  • #5
tiny-tim said:
the volume is a function of the radius (of the liquid surface) at time t, and the height at time t

the radius at time t is not 4, it depends the height :wink:

The maximum height being 16 and the radius 4 we can assume the radius y/4 or square(y)? is that it?
 
  • #6
yes, radius = y/4 :smile:
 
  • #7
tiny-tim said:
yes, radius = y/4 :smile:

So now I have volume = pi*y*((y/4)^2)/3

volume = pi*y^3/48

dV/dt = pi*y^2/48 dy/dt

-2 = pi*y^2/48 dy/dt

dy/dt = 96*y^2/pi

still incorrect :/ I'm doing this exercise for a friend. I don't remember the subject that well. I'm doing something wrong or something missing probably. Sorry for the incovinience
 
  • #8
ruiwp13 said:
volume = pi*y^3/48

dV/dt = pi*y^2/48 dy/dt

nooo :wink:
 
  • #9
tiny-tim said:
nooo :wink:

Is the formula that is incorrect? Because I don't have the formulas anymore... I'm helping a friend. I remembered it had something to do with related rates :p sorry to bother you

Or is it dV/dt = pi*3*y^2/48 dy/dt

Then -2 = pi*3*y^2/48 dy/dt

dy/dt = -96/pi*3*y^2

dy/dt = -32/pi*y^2 it's correct!

Thanks, a lot! Really helped me!
 
Last edited:
  • #10
ruiwp13 said:
Or is it dV/dt = pi*3*y^2/48 dy/dt

Yup! :biggrin:

(you knew that! :wink:)​
 
  • #11
tiny-tim said:
Yup! :biggrin:

(you knew that! :wink:)​

Really, thank you! I appreciated that you didn't give me the answer and helped me get there "by myself". Really nice forum.
 

Related to Differential Calculus - Related Rates

1. What is differential calculus?

Differential calculus is a branch of mathematics that deals with the study of rates of change of quantities. It involves the use of derivatives to determine how a function changes at a particular point.

2. What is related rates in differential calculus?

Related rates in differential calculus refers to the study of how the rates of change of two or more related variables are connected. It involves finding the rate of change of one variable with respect to another variable.

3. How is the chain rule used in related rates problems?

The chain rule is used in related rates problems to find the derivative of one variable with respect to another variable. This is necessary when the variables are not directly related, but are connected through a chain of intermediary variables.

4. What are some real-world applications of related rates in differential calculus?

Related rates in differential calculus has various real-world applications, such as predicting the rate of change of stock prices, determining the speed of an object's descent from a height, and calculating the rate of growth of a population.

5. How can I improve my understanding of related rates in differential calculus?

To improve your understanding of related rates in differential calculus, it is important to practice solving various related rates problems and familiarize yourself with the different techniques and formulas used. It is also helpful to study the underlying concepts and principles of differential calculus, such as derivatives and rates of change.

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