1. **State the problem:** A radioactive substance decays at a rate proportional to its mass. When the mass is 26 mg, the decay rate is 10 mg per week. We want to find a formula for the mass $m$ in terms of time $t$ (weeks). 2. **Set up the differential equation:** Since the decay rate is proportional to the mass, we write: $$\frac{dm}{dt} = -km$$ where $k > 0$ is the decay constant and the negative sign indicates decay. 3. **Use the given information to find $k$:** When $m = 26$ mg, the decay rate is 10 mg/week, so: $$\frac{dm}{dt} = -10 = -k \times 26$$ Solving for $k$: $$k = \frac{10}{26} = \frac{5}{13}$$ 4. **Solve the differential equation:** The general solution to $$\frac{dm}{dt} = -km$$ is $$m = m_0 e^{-kt}$$ where $m_0$ is the initial mass at $t=0$. 5. **Express the formula for $m$:** Since the problem does not specify $m_0$, we keep it general. The formula is: $$m = m_0 e^{-\frac{5}{13}t}$$ **Final answer:** $$\boxed{m = m_0 e^{-\frac{5}{13}t}}$$ This formula gives the mass $m$ after $t$ weeks, where $m_0$ is the initial mass.