1. **Problem 1: Investment Allocation** A person invested a total of 120000, part at 4% interest and the rest at 5%. The total interest earned is equivalent to 4.5% on the entire 120000. Find the amounts invested at each rate. 2. Let the amount invested at 4% be $x$. Then the amount invested at 5% is $120000 - x$. 3. The total interest from both investments is given by: $$0.04x + 0.05(120000 - x) = 0.045 \times 120000$$ 4. Simplify the equation: $$0.04x + 6000 - 0.05x = 5400$$ $$-0.01x + 6000 = 5400$$ $$-0.01x = 5400 - 6000 = -600$$ $$x = \frac{-600}{-0.01} = 60000$$ 5. So, $60000$ was invested at 4%, and the remainder $120000 - 60000 = 60000$ was invested at 5%. --- 1. **Problem 2: Demand Equation and Price Calculation** Given demand points: (60 units, 15.30 price) and (35 units, 19.30 price). Find the linear demand equation $P = mQ + b$. 2. Calculate slope $m$: $$m = \frac{19.30 - 15.30}{35 - 60} = \frac{4}{-25} = -0.16$$ 3. Find intercept $b$ using point (60, 15.30): $$15.30 = -0.16 \times 60 + b$$ $$15.30 = -9.6 + b$$ $$b = 15.30 + 9.6 = 24.9$$ 4. Demand equation: $$P = -0.16Q + 24.9$$ 5. Find price for: - 40 units: $$P = -0.16 \times 40 + 24.9 = -6.4 + 24.9 = 18.5$$ - 35 units: $$P = -0.16 \times 35 + 24.9 = -5.6 + 24.9 = 19.3$$ - 45 units: $$P = -0.16 \times 45 + 24.9 = -7.2 + 24.9 = 17.7$$ --- 1. **Problem 3: Break-even and Profit/Loss** Material cost per unit = 1.50, labor cost per unit = 5, fixed cost = 7000, wholesaler cost per unit = 8.20. 2. Total cost per unit: $$1.50 + 5 = 6.50$$ 3. Break-even point where total cost = total revenue: $$7000 + 6.50x = 8.20x$$ $$7000 = 8.20x - 6.50x = 1.70x$$ $$x = \frac{7000}{1.70} = 4117.65$$ So, at least 4118 units must be sold to make a profit. 4. Profit/Loss for 2500 units: Revenue = $8.20 \times 2500 = 20500$ Cost = $7000 + 6.50 \times 2500 = 7000 + 16250 = 23250$ Profit/Loss = $20500 - 23250 = -2750$ (loss) 5. Profit/Loss for 3000 units: Revenue = $8.20 \times 3000 = 24600$ Cost = $7000 + 6.50 \times 3000 = 7000 + 19500 = 26500$ Profit/Loss = $24600 - 26500 = -1900$ (loss) --- 1. **Problem 4: Market Equilibrium** Demand: $P = 90 - 3Q_D$ Supply: $P = 2 + 2Q_S$ 2. At equilibrium, quantity demanded equals quantity supplied: $Q_D = Q_S = Q$ 3. Set demand equal to supply: $$90 - 3Q = 2 + 2Q$$ $$90 - 2 = 3Q + 2Q$$ $$88 = 5Q$$ $$Q = \frac{88}{5} = 17.6$$ 4. Find equilibrium price: $$P = 90 - 3 \times 17.6 = 90 - 52.8 = 37.2$$ --- **Final answers:** - Problem 1: $60000$ at 4%, $60000$ at 5% - Problem 2: Demand equation $P = -0.16Q + 24.9$; prices: 40 units = 18.5, 35 units = 19.3, 45 units = 17.7 - Problem 3: Break-even at 4118 units; loss of 2750 at 2500 units; loss of 1900 at 3000 units - Problem 4: Equilibrium quantity = 17.6 units, price = 37.2