Results 1 to 3 of 3

Math Help - Differential Eqns

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    16

    Differential Eqns

    At time t, t \ge 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t = 0, the radius of the sphere is 1 and at t = 15, the radius is 2. (Volume of sphere is V = 4/3pir^3)

    a) Find the radius of the sphere as a fuction of t

    b) at what time t will the volume of the sphere be 27 times its volume @ t= 0?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Quote Originally Posted by ny_chow View Post
    At time t, t \ge 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t = 0, the radius of the sphere is 1 and at t = 15, the radius is 2. (Volume of sphere is V = 4/3pir^3)

    a) Find the radius of the sphere as a fuction of t

    b) at what time t will the volume of the sphere be 27 times its volume @ t= 0?
    a.) It's best to list know information first. We know that:
    \frac{dV}{dt}=\frac{k}{r} where k is a constant.
    V=\frac{4}{3}\pi r^3
    \frac{dV}{dr}=4\pi r^2

    \frac{dV}{dt} = \frac{dV}{dr}\cdot\frac{dr}{dt} \implies \frac{k}{r} = 4\pi r^2\cdot\frac{dr}{dt} \implies \frac{dr}{dt}=\frac{k}{4\pi r^3}

    Separate the variables and integrate: \int 4\pi r^3\,dr = \int k\,dt \implies \pi r^4 = kt+C.

    t=0, r=1 \implies \pi=C

    t=15, r=2 \implies 16\pi=15k+\pi \implies k=\pi

    So, \pi r^4 = \pi t+\pi \implies \boxed{r = \sqrt[4]{t+1}}

    I have to go now. If the second part is still unanswered when I get back, I'll help you out then.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2009
    Posts
    16
    i think i can do the second part on my own. it seems pretty straight forward. thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. non linear differential eqns population models
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: June 7th 2010, 10:56 PM
  2. Differential Eqns in Mathematica help please
    Posted in the Math Software Forum
    Replies: 0
    Last Post: April 8th 2010, 08:21 PM
  3. More Differential Eqns
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 4th 2009, 07:00 PM
  4. differential eqns ( try one please)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 10th 2008, 08:51 PM
  5. Replies: 1
    Last Post: December 5th 2007, 05:11 PM

Search Tags


/mathhelpforum @mathhelpforum