1. **Stating the problem:** We are given the function $$M(t) = 50 \left(\frac{1}{2}\right)^{\frac{t}{18.4}}$$ which models a quantity that decays over time, commonly used to represent half-life decay. 2. **Formula and explanation:** This is an exponential decay function where: - 50 is the initial amount at time $t=0$. - The base $\frac{1}{2}$ indicates the quantity halves every half-life period. - The exponent $\frac{t}{18.4}$ means the half-life period is 18.4 units of time. 3. **Important rules:** - When $t=0$, $M(0) = 50 \times 1 = 50$. - After one half-life ($t=18.4$), the quantity halves: $M(18.4) = 50 \times \left(\frac{1}{2}\right)^1 = 25$. - The function decreases exponentially as $t$ increases. 4. **Intermediate work example:** Calculate $M(36.8)$ (two half-lives): $$M(36.8) = 50 \left(\frac{1}{2}\right)^{\frac{36.8}{18.4}} = 50 \left(\frac{1}{2}\right)^2 = 50 \times \frac{1}{4} = 12.5$$ 5. **Summary:** This function models exponential decay with half-life 18.4. The quantity halves every 18.4 time units starting from 50.