Super Tough Engineering Maths Qn!

• bananaking

  4 Jun 09, 13:21

  Hey guys i have a enginnering maths 2 problem which needs help! this is really tough! please assist if possible! i cracked my brain for it but still can't solve it.

  1.THE PROBLEM

  A certain radioactive material is known to decay at a rate propotional to the amount present. A block of this substance having mass of 100g originally is observed. After 40 hours, it's mass reduces to 80g. Let m denote the mass of the radioactive material present at anytime t.

  a) Write down the differential equation which describes the rate of change m with respect to t.

  b)Solve the differential equation for m in terms of t.

  c)Find the mass of the substance after 80 hours.

  d)A decaying catalyst is added to the radioactive material after 80 hours. the radioactive material is now observed to decay at a rate proportional to the square of the amount of radioactive material present. If the mass of the radioactive material is 50g after 90 hours, find it's mass after 100 hours.

• wee_ws

  4 Jun 09, 14:20

  Hi,

    It is possible to proceed if we try to analyse the problem statement carefully.

    1. "decay at a rate proportional to the amount present" means:

         dm/dt = km, where k is the constant of proportionality.

    2. "having mass of 100g originally" means:

        m = 100 when t = 0.

    3. "After 40 hours, its mass reduces to 80g" means:

        m = 80 when t = 40.

    In part (a), we have dm/dt = km.

    In part (b), we solve the differential equation by the method of variable separable (i.e put all the m's on one side and all the t's on the other).

    integral 1/m dm = integral k dt

    ln |m| = kt + c

    |m| = e^(kt + c) = A e^(kt), where A = e^c

    m = plus/minus A e^(kt)

    m = B e^(kt), where B = plus/minus A.

    Now, we will use the information in points 2 and 3 to solve for B and then k, which I'll leave you to continue. Answers: B = 100, t = 1/80 ln (4/5).

    In (c), just find m when t = 80.

    In (d), apply the same analysis as in (a).

    Thanks!

  Cheers,
  Wen Shih

• bananaking

  5 Jun 09, 13:49

  woah! thanks a lot it really helps

• hisoka

  5 Jun 09, 14:14

  are you sure this is engineering problem? could have  appeared anywhere, including maths courses.

• bananaking

  5 Jun 09, 14:17

  yeah man.. my engine maths lecturer gave us this problem to solve as a holiday problem..

• wee_ws

  5 Jun 09, 16:32

  Hi,

    Most engineering maths modules cover calculus, vectors, complex numbers, linear algebra, probability and statistics. Thanks!

  Cheers,
  Wen Shih

• syncopation_music

  5 Jun 09, 23:31

  Originally posted by wee_ws:

  Hi,

    Most engineering maths modules cover calculus, vectors, complex numbers, linear algebra, probability and statistics. Thanks!

  Cheers,
  Wen Shih

  i begged to differ slightly from your view ....

   

  you have not include 3rd and 4th Order differentiation in Electric Circuits

  Laplace Transform ,Poisson and Normal Distribution , Newton Raphson method of Convergence and other additional maths

   

   

• wee_ws

  6 Jun 09, 07:45

  Hi,

    Thanks for adding more content to give completeness :)

  Cheers,
  Wen Shih

• bananaking

  9 Jun 09, 13:50

  hi can i know how to do you get the t from this?

  Now, we will use the information in points 2 and 3 to solve for B and then k, which I'll leave you to continue. Answers: B = 100, t = 1/80 ln (4/5).

  because i kind of get the different calculations

• deepak.c

  9 Jun 09, 14:07

   

  B = 100

  k = (1/40) (ln 4/5)

   

   

• wee_ws

  9 Jun 09, 15:54

  Originally posted by deepak.c:

   

  B = 100

  k = (1/40) (ln 4/5)

  Hi,

    Thanks for your correct answers :) I was careless.

  Cheers,
  Wen Shih

• ditzy

  9 Jun 09, 15:57

  Wow, this is the same question that came out for my o levels pure physics paper.

• eagle

  9 Jun 09, 16:09

  Originally posted by syncopation_music:

  i begged to differ slightly from your view ....

   

  you have not include 3rd and 4th Order differentiation in Electric Circuits

  Laplace Transform ,Poisson and Normal Distribution , Newton Raphson method of Convergence and other additional maths

   

   

  There's also Z-transform, Fourier Transform, Fourier Series, etc

   

   

  Originally posted by ditzy:

  Wow, this is the same question that came out for my o levels pure physics paper.

  

• deepak.c

  9 Jun 09, 19:16

  Originally posted by wee_ws:

  Hi,

    Thanks for your correct answers :) I was careless.

  Cheers,
  Wen Shih

   

  You're welcome.

  I help in any way I can.