1. **State the problem:** We have a firm with Fixed Cost (FC) = 1000 and Variable Cost (VC) = 50Q + 2Q^2. We need to write expressions for Total Cost (TC), Average Fixed Cost (AFC), Average Variable Cost (AVC), Average Total Cost (ATC), and Marginal Cost (MC). Then find the output levels where MC = AVC and MC = ATC. 2. **Write the cost expressions:** - Total Cost: $$TC = FC + VC = 1000 + 50Q + 2Q^2$$ - Average Fixed Cost: $$AFC = \frac{FC}{Q} = \frac{1000}{Q}$$ - Average Variable Cost: $$AVC = \frac{VC}{Q} = \frac{50Q + 2Q^2}{Q} = 50 + 2Q$$ - Average Total Cost: $$ATC = \frac{TC}{Q} = \frac{1000 + 50Q + 2Q^2}{Q} = \frac{1000}{Q} + 50 + 2Q = AFC + AVC$$ - Marginal Cost: $$MC = \frac{d(TC)}{dQ} = \frac{d}{dQ}(1000 + 50Q + 2Q^2) = 50 + 4Q$$ 3. **Find output where MC = AVC:** Set $$MC = AVC$$ $$50 + 4Q = 50 + 2Q$$ Subtract 50 from both sides: $$\cancel{50} + 4Q - \cancel{50} = \cancel{50} + 2Q - \cancel{50}$$ $$4Q = 2Q$$ Subtract 2Q from both sides: $$4Q - 2Q = 2Q - 2Q$$ $$2Q = 0$$ Divide both sides by 2: $$\frac{\cancel{2}Q}{\cancel{2}} = \frac{0}{2}$$ $$Q = 0$$ 4. **Find output where MC = ATC:** Set $$MC = ATC$$ $$50 + 4Q = \frac{1000}{Q} + 50 + 2Q$$ Subtract 50 from both sides: $$\cancel{50} + 4Q - \cancel{50} = \frac{1000}{Q} + \cancel{50} + 2Q - \cancel{50}$$ $$4Q = \frac{1000}{Q} + 2Q$$ Subtract 2Q from both sides: $$4Q - 2Q = \frac{1000}{Q} + 2Q - 2Q$$ $$2Q = \frac{1000}{Q}$$ Multiply both sides by Q: $$2Q \times Q = \frac{1000}{Q} \times Q$$ $$2Q^2 = 1000$$ Divide both sides by 2: $$\frac{2Q^2}{2} = \frac{1000}{2}$$ $$Q^2 = 500$$ Take square root: $$Q = \sqrt{500} = 10\sqrt{5} \approx 22.36$$ **Final answers:** - $$TC = 1000 + 50Q + 2Q^2$$ - $$AFC = \frac{1000}{Q}$$ - $$AVC = 50 + 2Q$$ - $$ATC = \frac{1000}{Q} + 50 + 2Q$$ - $$MC = 50 + 4Q$$ - Output where $$MC = AVC$$: $$Q = 0$$ - Output where $$MC = ATC$$: $$Q = 10\sqrt{5} \approx 22.36$$