1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derivatives, rates of change (trapezoidal prism)

  1. Dec 2, 2013 #1
    1. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m^3/min, how fast is the water level rising when the water is 30 cm deep?

    2. Relevant equations
    $$V=\frac{(a+b)h}{2}×L$$
    3. The attempt at a solution

    Given (I converted all to cm):
    $$a=30$$ $$\frac{da}{dt}=0$$ $$base=80$$ $$height=50$$ $$\frac{dV}{dt}=0.2$$ $$h=30$$ $$L=10$$ $$\frac{dL}{dt}=0$$

    I need to find $$\frac{dh}{dt}$$

    After I find the derivative of V using natural log.
    $$lnV=\frac{1}{2}ln(a+b)hL$$
    $$lnV=\frac{1}{2}ln(a+b)+lnh+lnL$$
    $$\frac{dV}{dt}=\frac{1}{2}(\frac{db/dt}{a+b}+\frac{dh/dt}{h})×V$$

    I don't know how I'm supposed to find $$\frac{db}{dt}$$ i get b=60.

    Any help is much appreciated.
     

    Attached Files:

    Last edited: Dec 2, 2013
  2. jcsd
  3. Dec 2, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    This is the same as the other one - you don't need to deal with all those logs.

    (a+b)L/2 is just a constant that you are given. Put k=2/(a+b)L
    Lynchpin You need to express b in terms of h.

    Think it through - let a is the bottom, and c is the top, the overall height is H.
    0<h<H
    when h=0, what is b(h) equal to?
    when h=H, what is b(h) equal to?
    what formula makes that happen? b(h)=

    * Rearrange your formula to make h the subject.
    * differentiate both sides to get dh/dt in terms of dV/dt.
    * You are given dV/dt.
     
    Last edited: Dec 2, 2013
  4. Dec 2, 2013 #3
    Thanks, but I still get this.

    $$2V=(a+b)hL$$ $$h=\frac{2V}{(a+b)L}=Vk$$ $$\frac{dh}{dt}=\frac{dV}{dt}k=0.2×\frac{2}{(30+60)1000}$$

    I don't get the right answer, which is 10/3.
     
  5. Dec 2, 2013 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Still the wrong formula.
    I misread you - I thought you put b as the top width - my bad.
    I have rewritten post #2 to reflet this.

    Think it through - let a is the bottom, and c is the top, the overall height is H.
    0<h<H
    when h=0, what is b(h) equal to?
    when h=H, what is b(h) equal to?
    what formula makes that happen? b(h)=
     
  6. Dec 2, 2013 #5

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Still the wrong formula.
    I misread you - I thought you put b as the top width - my bad.
    I have rewritten post #2 to reflet this.

    Think it through - let a is the bottom, and c is the top, the overall height is H.
    0<h<H
    when h=0, what is b(h) equal to?
    when h=H, what is b(h) equal to?
    what formula makes that happen? b(h)=
     
  7. Dec 2, 2013 #6

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Watch your units!

    If you are going to use cm for the dimensions of the trough, then the fill rate of 0.2 m^3/min should also be converted to units of cm^/min.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Derivatives, rates of change (trapezoidal prism)
Loading...