Skill: related-rates-optimization

Related Rates & Optimization Practice

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 Practice

Type your answer and press Check answer (or Enter). A wrong answer earns a hint; a second miss earns a stronger hint; a third miss walks you through the full solution. Scores and streaks are session-only.

Score 0 · Streak 0 · Problem 0

Score counts first-try correct answers. Streak counts consecutive correct answers.

Problem

Enter numbers (decimals ok). For “in terms of π” questions enter just the coefficient k. Round only when asked.

2 What you’ll practice

Every problem is generated fresh from templates across several question types — a problem never repeats until its whole pool is used up.

Sliding ladder

Pythagoras + implicit differentiation; mind the sign.

Ex: 10-ft ladder, dx/dt = 2 → dy/dt = ?

Inflating sphere

dV/dt from dr/dt via V = 4/3πr³.

Ex: r = 3, dr/dt = 2 → k = ? (in kπ)

Filling cone

Eliminate r with r = h/2 first.

Ex: h = 6, dh/dt = 2 → k = ? (in kπ)

Expanding ripple

dA/dt from dr/dt.

Ex: r = 10, dr/dt = 3 → k = ? (in kπ)

Max-volume box

V = x(s−2x)²; optimum at x = s/6.

Ex: 12×12 sheet → x = ?, Vmax = ?

Pen against a river

Three-sided fence; vertex of a parabola.

Ex: 40 m fence → x = ?, Amax = ?

Verify max vs min

Candidates vs. endpoints — who wins?

Ex: values 3, 11, 4, 9 → abs max at?

3 Watch out for these

The mistakes students make most often on these problems.

Differentiating before relating

Geometry first: write the equation (x²+y²=L², V=4/3πr³), then differentiate with respect to time.

Skipping the verification

Critical points are candidates — compare values at critical points and endpoints before crowning a max or min.

Dropping the sign

A decreasing quantity has a negative rate. dy/dt = −1.5 ft/s, not +1.5.

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