Answer to Question #292624 in Differential Equations for Ability

Question

A certain radioactive material is known to decay at a rate proportional to the amount

present. If the initially there is 50 milligrams of the material present and after two

hours it is observed that the material has lost 10% of its original mass. Find

a) An expression for the mass of the material remaining at any time t

b) The mass of the material after 4 hours

c) The time rate at which the material has decayed to one half of its initial mass

Answer 1

Let $A(t)=A=$ the amount of radioactive material.

Given $\dfrac{dA}{dt}=-kA,$ where $k$ is the proportionality constant.

$$\dfrac{dA}{A}=-kdt$$

Integrate

$$\int\dfrac{dA}{A}=-\int kdt$$

Let $A(0)=A_0=$ the initialamount of radioactive material.

$$A=A_0e^{-kt}$$

Given $A_0=50\ mg, A(2)=0.9A_0$

Substitute

$$0.9A_0=A_0e^{-k(2)}$$

$$-2k=\ln(0.9)$$

$$k=-\dfrac{1}{2}\ln (0.9) \ h^{-1}$$

a)

$$A(t)=50e^{\frac{1}{2}\ln (0.9)t}$$

$$A(t)=50(0.9)^{t/2}$$

b)

$$A(4)=50(0.9)^{4/2}$$

$$A(4)=40.5\ mg$$

c)

$$A(t_1)=\dfrac{1}{2}A_0$$

$$\dfrac{1}{2}A_0=A_0(0.9)^{t_1/2}$$

$$(0.9)^{-t_1/2}=2$$

$$t_1=-\dfrac{2\ln2}{\ln 0.9}$$

$$t_1\approx13.1576\ hour$$