Optimization Example

Calculus 1 · Axiom Academy

Maximizing the volume of an open-top box under a fixed surface-area constraint. An open-top box has a square base of side length x and height h . The total surface area (base plus four sides) is fixed at 600 square units. Find the dimensions that maximize the volume . Volume rises to a single peak, then falls — the maximum is the interior critical point. Nice work. You turned a constrained word problem into a single-variable max — the heart of every optimization problem. Objective + constraint: name the quantity to optimize ( V = x^2 h ) and the restriction that ties the variables together ( x^2 + 4xh = 600 ). Reduce to one variable: solve the constraint for h and substitute, giving . Critical points: gives x^2=200 , so . Verify the max: , so this critical point is a maximum — with and . The same four moves — objective, constraint, reduce, optimize — solve box, fence, can, and cost problems alike.

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