Final answer:
To find the maximum revenue along the demand curve Qd=90-2P, calculate the midpoint where the price elasticity of demand equals -1. The maximum revenue is $1,012.50, occurring when the price is $22.50, and the quantity demanded is 45 units.
Step-by-step explanation:
The student's question concerns finding the maximum revenue on a demand curve represented by the equation Qd=90-2P. To find the maximum revenue, we need to understand that revenue is calculated by multiplying the price per unit (P) by the quantity demanded (Qd). In a linear demand curve such as the one given (Qd=90-2P), the maximum revenue occurs at the midpoint of the demand curve. This is because at the midpoint, the price elasticity of demand is equal to -1, which is where total revenue is maximized.
The revenue function would be R = P × Qd, which transforms into R = P × (90 - 2P). This can further be written as R = 90P - 2P². To find the maximum value of the revenue function, we can take the derivative of R with respect to P and set it equal to zero. This gives 90 - 4P = 0. Solving for P gives us P = $22.50. At this price, the quantity demanded would be Qd = 90 - 2 × 22.50, which equals 45 units. Therefore, the maximum revenue would be R = 22.50 × 45, or $1,012.50.If using graphs rather than solving algebraically, the same outcome can be achieved by finding the vertex of the parabolic revenue curve, which also demonstrates where total revenue is at its peak.