Related Rates & Optimization Lab
How fast is the ladder sliding? What box holds the most? Related rates link changing quantities through implicit differentiation; optimization turns a constraint into a max-or-min hunt with the derivative tests.
1 Box optimization explorer
Cut squares of side x from an s×s sheet and fold up an open box. Drag the sliders: the curve is V(x), the gold marker is your box, and the dashed line is the true maximum at x = s/6.
V(x) = – in³
V′(x) = –
Optimum x = s/6 = –
Vmax = – in³
2 What to notice
- V(x) starts at 0, rises to a peak, and falls back to 0 at x = s/2 — both endpoints give flat sheets, not boxes.
- V′(x) is positive left of the peak and negative right of it: the first-derivative sign test in action.
- The peak sits at exactly x = s/6 for every sheet size — a pattern you can prove once and use forever.