Question:

A radioactive substance starts with an initial mass of 100 grams and decays according to the differential equation:

where $m$ is the mass of the substance at time $t$ (in years), and $k$ is the decay constant. Separate the variables and solve the differential equation to find the equation for the mass of the substance as a function of time. Given that the decay constant is $k = 0.02$ per year, find the mass of the substance after 5 years.

Answer:

To solve the given differential equation $\frac{dm}{dt} = -k \cdot m$, we can separate the variables and integrate.

Separating the variables:

Integrating both sides:

Using the fact that $\int\frac{1}{x}\,dx = \ln|x| + C_1$ and $\int -k\,dx = -kx + C_2$, the equation becomes:

Combining the constants of integration:

Exponentiating both sides:

Using the properties of logarithms, $e^{\ln|m| + C} = e^{\ln|m|} \cdot e^C = |m| \cdot e^C$, and $e^{-kt} = \frac{1}{e^{kt}}$, we get:

Since $e^C$ is a positive constant, we can replace it with $C_3$, another positive constant. Now, the equation becomes:

We can rewrite this equation as:

where $C_4 = \pm C_3$.

Given that the decay constant is $k = 0.02$ per year, and the initial mass of the substance is 100 grams, we can substitute these values into our equation to find the mass after 5 years:

Simplifying:

Since we know the initial mass is 100 grams, we can set $C_4 = 100$. Therefore:

Calculating this value:

So, the mass of the substance after 5 years is approximately 90.483 grams.

Therefore, the mass of the substance as a function of time is given by the equation $m = \frac{100}{e^{0.02t}}$, and after 5 years, the mass is approximately 90.483 grams.