# Math Help - constants of D.e

1. ## 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

2. Originally Posted by i_zz_y_ill
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:

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

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