Bread Profit 763482

1. **State the problem:** We have revenue function $$R(q) = 63q - 0.2q^2$$ and cost function $$C(q) = 20 + 17q$$ for $q$ loaves of bread. We want to find: a. The profit function $P(q)$. b. The production level $q$ that maximizes profit. c. The maximum profit. 2. **Find the profit function $P(q)$:** Profit is revenue minus cost: $$P(q) = R(q) - C(q)$$ Substitute the given functions: $$P(q) = (63q - 0.2q^2) - (20 + 17q)$$ Simplify: $$P(q) = 63q - 0.2q^2 - 20 - 17q$$ $$P(q) = (63q - 17q) - 0.2q^2 - 20$$ $$P(q) = 46q - 0.2q^2 - 20$$ 3. **Find the production level $q$ that maximizes profit:** Since $P(q)$ is a quadratic function with a negative coefficient for $q^2$, it opens downward and has a maximum at its vertex. The vertex of $$P(q) = -0.2q^2 + 46q - 20$$ is at $$q = -\frac{b}{2a}$$ where $a = -0.2$ and $b = 46$. Calculate: $$q = -\frac{46}{2 \times (-0.2)} = -\frac{46}{-0.4} = 115$$ So, the profit is maximized when producing $$q = 115$$ loaves. 4. **Find the maximum profit:** Substitute $q=115$ into $P(q)$: $$P(115) = 46(115) - 0.2(115)^2 - 20$$ Calculate each term: $$46 \times 115 = 5290$$ $$115^2 = 13225$$ $$0.2 \times 13225 = 2645$$ So, $$P(115) = 5290 - 2645 - 20 = 2625$$ **Final answers:** - a. $$P(q) = 46q - 0.2q^2 - 20$$ - b. $$q = 115$$ loaves - c. Maximum profit $$P(115) = 2625$$