1. **State the problem:** We have a linear demand function with two points: price $p=220$ when quantity demanded $q=40$, and price $p=240$ when $q=30$. The total cost function is given (though not explicitly stated in the prompt, we assume it is $TC(q)$). We need to find the maximum profit. 2. **Find the demand function:** The demand function is linear: $p = m q + b$. Using points $(40,220)$ and $(30,240)$: $$m = \frac{240 - 220}{30 - 40} = \frac{20}{-10} = -2$$ Using point $(40,220)$ to find $b$: $$220 = -2 \times 40 + b \Rightarrow b = 220 + 80 = 300$$ So demand function is: $$p = 300 - 2q$$ 3. **Express total revenue (TR):** $$TR = p \times q = (300 - 2q)q = 300q - 2q^2$$ 4. **Express total cost (TC):** The problem states total cost is given but does not provide the explicit formula. Since the problem is incomplete, we assume a typical quadratic cost function for demonstration: $$TC = 20q + 10q^2$$ 5. **Profit function:** $$\pi = TR - TC = (300q - 2q^2) - (20q + 10q^2) = 300q - 2q^2 - 20q - 10q^2 = 280q - 12q^2$$ 6. **Maximize profit:** Take derivative of $\pi$ with respect to $q$ and set to zero: $$\frac{d\pi}{dq} = 280 - 24q = 0 \Rightarrow 24q = 280 \Rightarrow q = \frac{280}{24} = 11.67$$ 7. **Calculate maximum profit:** $$\pi(11.67) = 280 \times 11.67 - 12 \times (11.67)^2 = 3267.6 - 1633.3 = 1634.3$$ Since the problem's options are in millions and the calculated profit is not matching, the exact total cost function is needed to finalize the answer. **Note:** Without the explicit total cost function, the exact maximum profit cannot be determined. **Summary:** - Demand function: $p = 300 - 2q$ - Revenue: $TR = 300q - 2q^2$ - Profit maximized at $q = \frac{280}{24} = 11.67$ (assuming $TC = 20q + 10q^2$) - Maximum profit approximately $1634.3$ (units depend on cost function) Please provide the total cost function to compute the exact maximum profit.