1. **State the problem:** We need to find the time $t$ (in hours) after Elizabeth takes her medicine when the amount of medicine in her blood, modeled by $M(t) = -t^2 + 12t$, is at its highest. 2. **Identify the type of function:** $M(t)$ is a quadratic function with a negative leading coefficient ($-1$), so its graph is a parabola opening downward. The maximum value occurs at the vertex. 3. **Formula for the vertex of a parabola:** For $f(t) = at^2 + bt + c$, the vertex $t$-coordinate is given by $$t = -\frac{b}{2a}$$ 4. **Apply the formula:** Here, $a = -1$ and $b = 12$, so $$t = -\frac{12}{2 \times (-1)} = -\frac{12}{-2} = 6$$ 5. **Interpretation:** The amount of medicine in Elizabeth's blood is highest 6 hours after taking the medicine. 6. **Check options:** The closest option to 6 hours is not listed exactly, but since 5 and 10 are options, the correct answer is not exactly given. The maximum is at 6 hours. **Final answer:** The amount of medicine is highest at $\boxed{6}$ hours after taking the medicine.