The Snow Cone Stand Math

9709 · 1.1 Quadratics · where the math is actually used

The Snow Cone Stand

You are running a snow cone stand. You pick the price. Everything else follows, and it follows a parabola. Find the price that makes the most money, then push your costs up until the whole thing stops being possible.

The setup

Charge too little and you barely cover your costs. Charge too much and nobody buys. Somewhere in between is the best price, and completing the square finds it exactly.

cones sold = 180 − 20 × price
Every dollar you add loses you 20 customers.
$1 per cone
Ice and syrup, for each cone you actually sell.
fixed cost
The stand and the ice machine. You pay it whether you sell any or not.
price you charge per cone

The curve is your profit at every possible price. The horizontal line is breaking even.

What the till says
Cones sold
Money taken in
Ice and syrup
Stand and machine
Profit
The math behind it

Why is it a parabola?

Nobody handed you this equation. It comes out of the stand itself, and the p² appears the moment you multiply price by how many you sell.

1Money in is price times cones: p(180 − 20p)
2There it is. A price times something containing a price gives −20p².
3Ice and syrup cost 1 × (180 − 20p), and the stand costs C whatever happens.
4Profit is what is left: (p − 1)(180 − 20p) − C
5Multiply out: −20p² + 200p − (180 + C)

a is negative, so the curve opens downward and the vertex is the highest point. That has to be right: profit climbs, peaks, then falls again once you price yourself out.

What Δ decides here

Breaking even means profit is zero, so the break-even prices are the roots. How many there are is exactly what the discriminant counts, and here it answers a real question: can this stand work at all?

Two break-even prices, and profit between themΔ > 0
One price that exactly breaks even, and nothing betterΔ = 0
No price works. You lose money whatever you charge.Δ < 0

Push the fixed cost up and watch the two break-even points slide toward each other. They meet at $320, and past that the curve never reaches the line. That is the same moment the top of the parabola touches zero, because the highest profit you can reach is 320 − C.