Number and Algebra
Functions
Geometry and Trigonometry
Statistics and Probability
Calculus
Write the differential equation that models the decay of a material X(t)X(t)X(t) if its decay rate is inversely proportional to the current amount XXX.
Solve the differential equation dXdt=−kX\dfrac{dX}{dt}=-\dfrac{k}{X}dtdX=−Xk subject to the initial condition X(0)=X0 X(0)=X_0\,X(0)=X0.
Given the solution X(t)=X02−2ktX(t)=\sqrt{X_0^2-2kt}X(t)=X02−2kt, find the time TTT at which X(T)=0X(T)=0X(T)=0.
If X(0)=10 kgX(0)=10\text{ kg}X(0)=10 kg and k=3 kg2/dayk=3\text{ kg}^2/\text{day}k=3 kg2/day in the model dXdt=−kX\dfrac{dX}{dt}=-\dfrac{k}{X}dtdX=−Xk, how many days until XXX reaches zero?
For the same model with X(0)=5X(0)=5X(0)=5 and k=2k=2k=2, find the time when X=1X=1X=1.
Express the time ttt as a function of XXX for the decay model dX/dt=−k/XdX/dt=-k/XdX/dt=−k/X with X(0)=X0 X(0)=X_0\,X(0)=X0.
A sample decays according to dX/dt=−k/XdX/dt=-k/XdX/dt=−k/X. If its half-life is t1/2=4t_{1/2}=4t1/2=4, express kkk in terms of X0X_0X0.
Show that the time t1/2t_{1/2}t1/2 for X(t)X(t)X(t) to decay to half its initial amount X0X_0X0 is t1/2=3X028kt_{1/2}=\dfrac{3X_0^2}{8k}t1/2=8k3X02 if the model is dX/dt=−k/XdX/dt=-k/XdX/dt=−k/X.
Assume dX/dt=−k/XdX/dt=-k/XdX/dt=−k/X and that X(0)=X0X(0)=X_0X(0)=X0. Derive the general time trt_rtr for X(tr)=rX0X(t_r)=rX_0X(tr)=rX0, where 0<r<10<r<10<r<1.
A pollutant follows dP/dt=−k/PdP/dt=-k/PdP/dt=−k/P with P(0)=50P(0)=50P(0)=50. After 8 days P=10P=10P=10. Determine kkk and the extinction time TTT when P=0P=0P=0.
A reactor’s mass MMM follows dM/dt=−k/MdM/dt=-k/MdM/dt=−k/M and initially M(0)=100M(0)=100M(0)=100. Find the time to reduce MMM to 20 and comment on whether the model allows negative mass.
Explain why the solution X(t)=X02−2ktX(t)=\sqrt{X_0^2-2kt}X(t)=X02−2kt implies a finite extinction time and contrast this with exponential decay models.
Previous
No previous topic
Next
Question Type 2: Separating a differential equation into two variables