1. **State the problem:** We are given the power function $$P(I) = -5I^2 + 100I$$ where $I$ is the current in amperes. We need to find the current $I$ that maximizes the power and then find the maximum power. 2. **Formula and rules:** To find the maximum of a quadratic function $$P(I) = aI^2 + bI + c$$ where $a < 0$, the vertex formula gives the $I$ value that maximizes $P$ as $$I = -\frac{b}{2a}$$. 3. **Identify coefficients:** Here, $a = -5$, $b = 100$, and $c = 0$. 4. **Calculate the current that maximizes power:** $$I = -\frac{100}{2 \times (-5)} = -\frac{100}{-10} = 10$$ 5. **Find the maximum power by substituting $I=10$ into $P(I)$:** $$P(10) = -5 \times 10^2 + 100 \times 10 = -5 \times 100 + 1000 = -500 + 1000 = 500$$ 6. **Summary:** The current that maximizes power is $10$ amperes, and the maximum power produced is $500$ watts.