This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The transformation from the demand function to the Marginal Revenue (MR) function involves several steps: Step 1: Convert the demand function to the inverse demand function. The given demand function is: QD = 400 - 0.5P To find the inverse demand function, we need to express Price (P) in terms of Quantity Demanded (QD). Let's use Q for QD. Q = 400 - 0.5P Rearrange to isolate P: 0.5P = 400 - Q Multiply both sides by 2 to solve for P: P = 2(400 - Q) P = 800 - 2Q This is the inverse demand function. Step 2: Derive the Total Revenue (TR) function. Total Revenue (TR) is calculated as Price (P) multiplied by Quantity (Q): TR = P × Q Substitute the inverse demand function for P: TR = (800 - 2Q)Q TR = 800Q - 2Q^2 Step 3: Derive the Marginal Revenue (MR) function. Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to Quantity (Q): MR = (dTR)/(dQ) MR = (d)/(dQ)(800Q - 2Q^2) Differentiate each term: (d)/(dQ)(800Q) = 800 (d)/(dQ)(-2Q^2) = -2 × 2Q^2-1 = -4Q So, the Marginal Revenue function is: MR = 800 - 4Q This shows how QD = 400 - 0.5P becomes MR = 800 - 4Q. What's next? 📸