1. **State the problem:** We want to find the maximum profit and the number of units $x$ that must be produced and sold to achieve this maximum profit. 2. **Given functions:** - Revenue: $$R(x) = 60x - 0.5x^2$$ - Cost: $$C(x) = 5x + 15$$ 3. **Profit function:** Profit $$P(x)$$ is revenue minus cost: $$P(x) = R(x) - C(x) = (60x - 0.5x^2) - (5x + 15) = 60x - 0.5x^2 - 5x - 15$$ Simplify: $$P(x) = 55x - 0.5x^2 - 15$$ 4. **Find the maximum profit:** To find the maximum, take the derivative of $$P(x)$$ and set it to zero: $$P'(x) = 55 - x = 0$$ Solve for $$x$$: $$x = 55$$ 5. **Check the second derivative to confirm maximum:** $$P''(x) = -1 < 0$$ which means the profit function is concave down and $$x=55$$ is a maximum. 6. **Calculate maximum profit:** Substitute $$x=55$$ into $$P(x)$$: $$P(55) = 55(55) - 0.5(55)^2 - 15 = 3025 - 0.5(3025) - 15 = 3025 - 1512.5 - 15 = 1497.5$$ **Final answer:** - Maximum profit is $$1497.50$$ dollars. - Number of units to produce and sell is $$55$$ units.