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Math Help - constants of D.e

  1. #1
    Member i_zz_y_ill's Avatar
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    constants of D.e

    If any one has the time to look atthis that would be great. Im stuck on (iv)

    Q A radioactive element X decays into he radioactive element Y which decays into the stable eement Z. Thedecay can be modelled by th differential equations:
    (1) dx/dt = -0.1X
    (2)dy/dt=0.1x-0.2y
    (3)dz/dt=0.2Y
    Where x.y and z are the masses (in milligrams) of X,Y AND Z respectively at time t seconds. When t=0, the mass of X is 25mg, and there is no or Z.
    (i) Solve eqn (1) to find x in terms of t,Sketch graph of your solution(DONE x=25e^-0.1t)
    (ii)se your solution for x to solve (2) to find y in terms of t. (DONE y=25(e^-0.1t-e^-0.2t))
    (iii)Calculatethe range of times for which the amount of Y is increasng. Sketch a graph of the mass of Y against time, showing the maximum alue. (DONE T IS SMALLER THAT 10LN2)
    (iv)*Without solving eqn (3), show that x+y+z is constant. Hence find z in terms of t, and verify this satidfies eqn(3) and the initial conditions.

    for (iv) the mark scheme does dx/dt+dy/dt+dz/dt=0 which implies a constant which I sortov understand but then in the mark scheme it requires a mar to get the constant as 25 i.e x+y+z=25 and I have no idea how to get this or find z in terms of t, the very last bit verifying should be easy. Does any one know how? thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by i_zz_y_ill View Post
    If any one has the time to look atthis that would be great. Im stuck on (iv)

    Q A radioactive element X decays into he radioactive element Y which decays into the stable eement Z. Thedecay can be modelled by th differential equations:
    (1) dx/dt = -0.1X
    (2)dy/dt=0.1x-0.2y
    (3)dz/dt=0.2Y
    Where x.y and z are the masses (in milligrams) of X,Y AND Z respectively at time t seconds. When t=0, the mass of X is 25mg, and there is no or Z.
    (i) Solve eqn (1) to find x in terms of t,Sketch graph of your solution(DONE x=25e^-0.1t)
    (ii)se your solution for x to solve (2) to find y in terms of t. (DONE y=25(e^-0.1t-e^-0.2t))
    (iii)Calculatethe range of times for which the amount of Y is increasng. Sketch a graph of the mass of Y against time, showing the maximum alue. (DONE T IS SMALLER THAT 10LN2)
    (iv)*Without solving eqn (3), show that x+y+z is constant. Hence find z in terms of t, and verify this satidfies eqn(3) and the initial conditions.

    for (iv) the mark scheme does dx/dt+dy/dt+dz/dt=0 which implies a constant which I sortov understand but then in the mark scheme it requires a mar to get the constant as 25 i.e x+y+z=25 and I have no idea how to get this or find z in terms of t, the very last bit verifying should be easy. Does any one know how? thanks
    for iv consider:

    \frac{d}{dt}(x+y+z)=\frac{dx}{dt}+\frac{dy}{dt}+\f  rac{dz}{dt}=<br />
(-0.1x)+(0.1x-0.2y)+(0.2y)=0

    and since at t=0\ x=25,\ y=0,\ z=0,\ x(0)+y(0)=z(0)=25, and as x+y+z is constant x+y+z=25 at all times.

    You have expressions for x and y in terms of t, so to find z in terms of t just put z(t)=25-x(t)-y(t).


    RonL
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