1. **State the problem:** We are given the daily production cost function for manufacturing $x$ chairs as $$C(x) = 50 + 14x + \frac{x^2}{2}$$ and the demand function as $$p = 150 - \frac{3x}{2}$$. We need to find the marginal cost and marginal revenue when the demand is 8 units. 2. **Recall formulas:** - Marginal cost (MC) is the derivative of the cost function with respect to $x$: $$MC = C'(x)$$. - Marginal revenue (MR) is the derivative of the revenue function with respect to $x$. Revenue $R(x)$ is price times quantity: $$R(x) = p \times x$$. 3. **Find $x$ when demand is 8 units:** Given demand $p = 8$, use the demand function: $$8 = 150 - \frac{3x}{2}$$ Solve for $x$: $$\frac{3x}{2} = 150 - 8 = 142$$ $$x = \frac{142 \times 2}{3} = \frac{284}{3} \approx 94.67$$ 4. **Calculate marginal cost:** Differentiate $C(x)$: $$C'(x) = \frac{d}{dx} \left(50 + 14x + \frac{x^2}{2}\right) = 0 + 14 + x = 14 + x$$ Evaluate at $x = \frac{284}{3}$: $$MC = 14 + \frac{284}{3} = \frac{42}{3} + \frac{284}{3} = \frac{326}{3} \approx 108.67$$ 5. **Calculate marginal revenue:** First, express revenue: $$R(x) = p \times x = \left(150 - \frac{3x}{2}\right) x = 150x - \frac{3x^2}{2}$$ Differentiate $R(x)$: $$R'(x) = 150 - 3x$$ Evaluate at $x = \frac{284}{3}$: $$MR = 150 - 3 \times \frac{284}{3} = 150 - 284 = -134$$ **Final answers:** - Marginal cost when demand is 8 units: $$\approx 108.67$$ - Marginal revenue when demand is 8 units: $$-134$$