1. **Problem statement:** A company produces units with a linear total cost function. It can produce 1,000 units at a total cost of 300000 and 2,000 units at a total cost of 400000. Units sell for 180 each. We need to find the revenue, cost, and profit functions, and identify fixed and variable costs. 2. **Formulas and rules:** - Total cost function: $$C(x) = mx + b$$ where $m$ is variable cost per unit and $b$ is fixed cost. - Revenue function: $$R(x) = p \times x$$ where $p$ is price per unit. - Profit function: $$P(x) = R(x) - C(x)$$ 3. **Find variable cost $m$ and fixed cost $b$:** Given: $$C(1000) = 300000$$ $$C(2000) = 400000$$ Set up equations: $$m \times 1000 + b = 300000$$ $$m \times 2000 + b = 400000$$ Subtract first from second: $$m \times 2000 + b - (m \times 1000 + b) = 400000 - 300000$$ $$1000m = 100000$$ $$m = \frac{100000}{1000} = 100$$ Substitute $m=100$ into first equation: $$100 \times 1000 + b = 300000$$ $$100000 + b = 300000$$ $$b = 300000 - 100000 = 200000$$ 4. **Write cost function:** $$C(x) = 100x + 200000$$ 5. **Write revenue function:** Price per unit $p = 180$ $$R(x) = 180x$$ 6. **Write profit function:** $$P(x) = R(x) - C(x) = 180x - (100x + 200000) = 80x - 200000$$ 7. **Summary:** - Revenue function: $$R(x) = 180x$$ - Cost function: $$C(x) = 100x + 200000$$ - Profit function: $$P(x) = 80x - 200000$$ - Fixed cost: 200000 - Variable cost per unit: 100