The mass, $m$ grams, of a radioactive substance, $t$ years after first being observed, is modelled by the equation $$m = 25e^{0.05t}$$ According to the model, a) find the mass of the radioactive substance six months after it was first observed, b) show that $rac{dm}{dt} = km$, where $k$ is a constant to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2017 - Paper 2

To show that rac{dm}{dt} = km, we need to differentiate the equation for $m$ with respect to $t$:

Starting with:

$m = 25e^{0.05t}$

Using the chain rule, we find:

$\frac{dm}{dt} = 25 \cdot \frac{d}{dt}(e^{0.05t}) = 25 \cdot (0.05e^{0.05t})$

This simplifies to:

$\frac{dm}{dt} = 1.25e^{0.05t}$

Now, we can express $m$ in our equation:

Since $m = 25e^{0.05t}$, we can rewrite our derivative:

$\frac{dm}{dt} = 0.05m$

Thus, we can identify that $k = 0.05$, and therefore:

$\frac{dm}{dt} = km$

where $k = 0.05$. This proves the statement.