Revenue Optimization

Algebra 1 · Axiom Academy

Revenue Optimization: Finding the Sweet Spot Charge too little and you leave money on the table; too much and the crowd walks away. One curve finds the perfect price. Maya runs a lemonade stand . She's noticed the catch every shop owner faces: raise the price and fewer people buy; drop it and they flock back, but each cup earns less. Her survey says about 50 people would buy if it were free, and she loses roughly 20 customers for every 1 she adds. What price makes her the most money? Drag the price and watch the stand. A higher price earns more per cup but thins the crowd; a lower price packs the line but barely pays. Revenue is price times cups — hunt for the setting that makes the bar tallest. Plot revenue against price and the tradeoff becomes a single hill: it climbs, tops out, and falls. Drag the price along the curve and find the very top — that's the most money the stand can make in a day. Eyeballing the peak is close, but algebra makes it exact. For R = -20p^2 + 50p the top sits at . Slide the guess line until it locks onto the vertex and watch the formula fill in. Maya's sweet spot is 1.25 a cup — about 25 customers a day for 31.25 , the most any price can earn. That's the move behind every pricing decision: when demand falls steadily with price, revenue is a downward parabola , and its peak is the vertex . The same math sets ticket prices, subscription tiers, and ad rates.

This is the written version of the interactive lesson above. See the full Algebra 1 course.