Final answer:
To keep weekly revenue constant as the demand for lemonade drops by 1 cup each week, raise the price by approximately 0.41 cents per cup each week. Initially, 40 cups are sold at 40 cents each, yielding $16 in weekly revenue. The calculation involves solving for the price increase needed to offset the decrease in quantity sold.
Step-by-step explanation:
To keep weekly revenue constant while demand is dropping, you first need to understand the initial revenue. Currently, you sell 40 cups at 40¢ each, which totals to $16 (40 cups × $0.40/cup). To maintain this weekly revenue of $16, you have to compensate for each cup lost in demand by increasing the price per cup accordingly.
Let's denote the number of cups sold per week as Q and the price per cup as P. Since demand falls by 1 cup each week and revenue R is constant, the equation R = P × Q implies that the rate of change of the price, dP/dt, must adjust so that dR/dt = d(P × Q)/dt = 0. Assume demand drops to 39 cups in the next week. Then, to find out by how much to increase the price, set up and solve the equation: 16 = (40¢ + dP) × (40 cups - 1 cup).
Adding 1 cent to the price per cup will result in $15.60 in revenue (39 cups × 41¢) rather than the required $16. So, you need to increase the price by more than 1 cent per cup. Solving the equation, dP = 16/39 - 40¢.
Therefore, you have to raise the price by 0.410256 cents (or approximately 0.41 cents) per cup each week to maintain the revenue of $16 as demand falls by 1 cup each week.