Results 1 to 5 of 5

Math Help - Integration, Maclaurin series, and radioactivity

  1. #1
    Junior Member
    Joined
    May 2008
    Posts
    55

    Integration, Maclaurin series, and radioactivity

    integrate:-

    1. integral (4x-x^2) ^ (1/2)from 0 to 4
    i know that I should use completing the square
    so i ended up with (-4-(x-2)^2)^(1/2)

    however, it doesn't look right to me, May I get a helpful solution?

    2.
    A certain radioactive isotope is observed to decay to 98% of its initial amount over a
    period of one year.
    a) Assume that the sample has an initial mass of 100g. Find a function that represents
    the mass of a sample as a function of time (in years).
    b) What is the half-life of the isotope?
    c) How long will it take the sample with initial mass of 100g to decay to a mass of 8g?


    3.
    Find the Maclaurin series for the functions sinh(
    x) and cosh(x) by using the Maclaurin
    series for
    ex and the de nitions of sinh(x) and cosh(x) in terms of ex. Compute the radius

    of convergence for each series.
    Last edited by Sally_Math; April 27th 2010 at 03:40 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Master Of Puppets
    pickslides's Avatar
    Joined
    Sep 2008
    From
    Melbourne
    Posts
    5,236
    Thanks
    28
    Quote Originally Posted by Sally_Math View Post
    integrate:-

    1. integral (4x-x^2) ^ (1/2) from 0 to 4
    i know that I should use completing the square
    so i ended up with (-4-(x-2)^2)^(1/2)

    however, it doesn't look right to me, May I get a helpful solution?


    This makes no sense to me, sorry.

    Quote Originally Posted by Sally_Math View Post
    2.
    A certain radioactive isotope is observed to decay to 98% of its initial amount over a
    period of one year.
    a) Assume that the sample has an initial mass of 100g. Find a function that represents
    the mass of a sample as a function of time (in years).
    b) What is the half-life of the isotope?
    c) How long will it take the sample with initial mass of 100g to decay to a mass of 8g?
    Try M = 100\times 0.98^t where t is years and M is mass.

    For half life sub M=50



    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2008
    Posts
    55
    integral (4x-x^2) ^ (1/2)from 0 to 4
    i know that I should use completing the square
    so i ended up with (-4-(x-2)^2))^(1/2)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,405
    Thanks
    1328
    Quote Originally Posted by Sally_Math View Post
    integral (4x-x^2) ^ (1/2)from 0 to 4
    i know that I should use completing the square
    so i ended up with (-4-(x-2)^2))^(1/2)
    No, that "4" cannot possibly be negative.

    4x- x^2= -(x^2- 4x). To 'complete the square' you must add and subtract (4/2)^2= 4: [tex]4x- x^2= -(x^2- 4x+ 4- 4)= -(-4)- (x- 2)^2= 4- (x- 2)^2.

    That is \int \sqrt{4x- x^2}dx= \int \sqrt{4- (x-2)^2}dx.

    Try a trig substitution.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,405
    Thanks
    1328
    Quote Originally Posted by Sally_Math View Post
    integrate:-

    1. integral (4x-x^2) ^ (1/2)from 0 to 4
    i know that I should use completing the square
    so i ended up with (-4-(x-2)^2)^(1/2)

    however, it doesn't look right to me, May I get a helpful solution?

    2.
    A certain radioactive isotope is observed to decay to 98% of its initial amount over a
    period of one year.
    a) Assume that the sample has an initial mass of 100g. Find a function that represents
    the mass of a sample as a function of time (in years).
    b) What is the half-life of the isotope?
    c) How long will it take the sample with initial mass of 100g to decay to a mass of 8g?

    (a) Let Q(t) be the quantity of isotope, in g, after t years. Saying that it decays at a steady rate means that \frac{dQ}{dt}= kQ for some (negative) number k. Rewrite that as \frac{dQ}{Q}= kdt and integrate both sides. Solving for Q will give an equation involving ln(Q) which gives Q as an exponential function of t involving the unknown value k and the integration constant C.

    Use the fact that Q(0)= 100 and Q(1)= .98(100)= 98 to find k and c.

    (b)The "half life" is the time it take to decay to half the original amount. Since the original amount was 100 g, half is 50 g. Set the function in (a) equal to 50 and solve for t.

    (c) Set the function you found in (a) equal to 8 and solve for t.

    3.
    Find the Maclaurin series for the functions sinh(
    x) and cosh(x) by using the Maclaurin
    series for
    ex and the de nitions of sinh(x) and cosh(x) in terms of ex. Compute the radius

    of convergence for each series.
    What are the definitions of sinh(x) and cosh(x) in terms of e^x?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: January 26th 2010, 08:06 AM
  2. Replies: 2
    Last Post: September 16th 2009, 07:56 AM
  3. Binomial Series to find a Maclaurin Series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 21st 2009, 07:15 AM
  4. Exponential Decay-Radioactivity
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 14th 2008, 07:00 PM
  5. Replies: 1
    Last Post: May 5th 2008, 09:44 PM

Search Tags


/mathhelpforum @mathhelpforum