Of course! Let’s break down the problem step by step. --- ## **Restating the Problem** - There is a bar with only **one cocktail bar**. - The bar has a **willingness to sell** (minimum price) and each student has a **willingness to buy** (maximum price). - **Willingness to buy** (students): - $25 at $20 per cocktail - $10 at $10 per cocktail - **Willingness to sell** (the bar): - 100 cocktails/day at $10 per cocktail - 250 cocktails/day at $15 per cocktail - **Q** is the quantity (number of cocktails sold per day). - **P** is the price per cocktail. --- ## **Question a: Demand Function (Qd as a function of P)** ### **Using the given data:** - At $20, 25 cocktails are demanded. - At $10, 100 cocktails are demanded. ### **Find the linear demand function:** The general form: \( Q_d = a - bP \) We have two points: - When \( P = 20, Q_d = 25 \) - When \( P = 10, Q_d = 100 \) Set up the equations: 1. \( 25 = a - b(20) \) 2. \( 100 = a - b(10) \) Subtract equation 1 from 2: \[ 100 - 25 = (a - b \cdot 10) - (a - b \cdot 20) \] \[ 75 = b(20 - 10) = 10b \] \[ b = 7.5 \] Plug \( b \) back into one equation: \[ 25 = a - 7.5 \times 20 \] \[ 25 = a - 150 \] \[ a = 175 \] **So, the demand function is:** \[ \boxed{Q_d = 175 - 7.5P} \] --- ## **Question b: Supply Function (Qs as a function of P)** ### **Given:** - At $10, 100 cocktails supplied. - At $15, 250 cocktails supplied. General form: \( Q_s = c + dP \) Two points: - \( P = 10, Q_s = 100 \) - \( P = 15, Q_s = 250 \) Set equations: 1. \( 100 = c + d(10) \) 2. \( 250 = c + d(15) \) Subtract equation 1 from 2: \[ 250 - 100 = (c + 15d) - (c + 10d) \] \[ 150 = 5d \] \[ d = 30 \] Plug \( d \) back in: \[ 100 = c + 30 \times 10 \] \[ 100 = c + 300 \] \[ c = -200 \] **So, the supply function is:** \[ \boxed{Q_s = -200 + 30P} \] --- ## **Question c: Find the Equilibrium Price and Quantity** Set \( Q_d = Q_s \): \[ 175 - 7.5P = -200 + 30P \] \[ 175 + 200 = 30P + 7.5P \] \[ 375 = 37.5P \] \[ P^* = 10 \] Now plug \( P^* \) into either equation to get \( Q^* \): \[ Q^* = 175 - 7.5 \times 10 = 175 - 75 = 100 \] (or \( Q^* = -200 + 30 \times 10 = -200 + 300 = 100 \)) **Equilibrium:** - **Price:** \(\boxed{10}\) - **Quantity:** \(\boxed{100}\) --- ## **Question d: Clearly label axes and plot supply and demand curves** **You would draw:** - X-axis: **Quantity of Cocktails (Q)** - Y-axis: **Price per Cocktail (P)** - **Demand curve:** downward sloping from (Q = 25, P = 20) to (Q = 100, P = 10) - **Supply curve:** upward sloping from (Q = 100, P = 10) to (Q = 250, P = 15) - **Equilibrium point:** where they cross, at (Q = 100, P = 10) --- ### **Summary Table** | Function | Equation | Intercepts (Q) | Slope | |---------------------|---------------------|------------------------|--------------| | Demand | Qd = 175 - 7.5P | Q = 175 (P = ) | -7.5 | | Supply | Qs = -200 + 30P | Q = -200 (P = ) | +30 | | **Equilibrium** | Q = 100, P = 10 | | | --- If you need help drawing the graph or further clarification, let me know!