1. **State the problem:** We have a manufacturing company with cost model $$C(x) = 0.35x^3 + 0.02x^2 + 4x + 240$$ and sales price model $$S(x) = 60 - 0.02x$$, where $x$ is the number of items sold in thousands. We need to: - Write an equation for revenue. - Write an equation for profit. - Find the total profit. 2. **Revenue formula:** Revenue is price times quantity sold, so $$R(x) = x \times S(x)$$ 3. **Write the revenue function:** Substitute $S(x)$: $$R(x) = x(60 - 0.02x)$$ 4. **Expand revenue:** $$R(x) = 60x - 0.02x^2$$ 5. **Profit formula:** Profit is revenue minus cost: $$P(x) = R(x) - C(x)$$ 6. **Substitute $R(x)$ and $C(x)$:** $$P(x) = (60x - 0.02x^2) - (0.35x^3 + 0.02x^2 + 4x + 240)$$ 7. **Simplify profit:** $$P(x) = 60x - 0.02x^2 - 0.35x^3 - 0.02x^2 - 4x - 240$$ Combine like terms: $$P(x) = -0.35x^3 - (0.02x^2 + 0.02x^2) + (60x - 4x) - 240$$ $$P(x) = -0.35x^3 - 0.04x^2 + 56x - 240$$ 8. **Interpretation:** The profit function in standard polynomial form is: $$\boxed{P(x) = -0.35x^3 - 0.04x^2 + 56x - 240}$$ This function models the company's profit based on thousands of items sold. 9. **Total profit:** To find total profit, we need a specific $x$ value. Since none is given, the profit function itself is the answer. --- **Summary:** - Revenue function: $$R(x) = 60x - 0.02x^2$$ - Profit function: $$P(x) = -0.35x^3 - 0.04x^2 + 56x - 240$$ This completes the problem.