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Related rates solver

Solve classic related-rates problems from a template: expanding sphere, spreading ripple, inflating balloon, sliding ladder, filling cone, shadow from a lamppost, two vehicles separating, and a changing rectangle. Pick a scenario, enter the known quantities and rates, and read off the unknown rate with its governing equation.

Related tools

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Bernoulli ODE SolverSolves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable.First-Order ODE SolverSolves dy/dx = f(x, y) from an initial condition with RK4, integrating outward in both directions, and plots the solution curve. Growth, decay, Newton cooling, logistic and separable cases are one click away.Second-Order ODE SolverSolves a·y″ + b·y′ + c·y = g(x) with two initial conditions. Gives the characteristic roots, names the regime (undamped, under-damped, critically damped, over-damped), derives a particular solution for constant, polynomial and exponential forcing, and plots the result.System of ODEs SolverSolves 2×2 and 3×3 linear systems x′ = Ax with eigenvalues, eigenvectors and the closed-form general solution, classifies the fixed point (node, spiral, centre, saddle, degenerate), and integrates arbitrary nonlinear 2D systems numerically with a phase portrait.Absolute Value Equation SolverSolve absolute-value equations like |x+3| = 5, 3|x+2| = 12 or |x+1| = |x−3| with step-by-step case analysis and exact solutions.Absolute Value Inequality SolverSolve absolute-value inequalities such as |x+3| < 5 or |x−2| > 3 and get the solution as an interval, with AND/OR reasoning explained step by step.Cubic Equation SolverSolve ax³ + bx² + cx + d = 0 for all three roots (real and complex), with the discriminant, the nature of the roots and steps.Exponential Equation SolverSolve exponential equations of six types — basic, coefficient, linear exponent, two bases, quadratic-in-a^x and shifted — with logarithms and a full step-by-step solution.

Related rates solver works straight from this page — free, instant, nothing to install. It sits under Calculus in our catalogue, alongside Bernoulli ODE Solver and First-Order ODE Solver.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Related rates solver do?

Solve classic related-rates problems from a template: expanding sphere, spreading ripple, inflating balloon, sliding ladder, filling cone, shadow from a lamppost, two vehicles separating, and a changing rectangle. Pick a scenario, enter the known quantities and rates, and read off the unknown rate with its governing equation.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

How is Related rates solver different from Bernoulli ODE Solver?

They sit next to each other but answer different questions. Bernoulli ODE Solver: solves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

First-Order ODE Solver is the closest one after this: Solves dy/dx = f(x, y) from an initial condition with RK4, integrating outward in both directions, and plots the solution curve. Growth, decay, Newton cooling, logistic and separable cases are one click away.

What else is worth having open alongside it?

Second-Order ODE Solver and System of ODEs Solver — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

All guides →
ExplainerThe Pigeonhole Principle Proves a Thing Exists Without Finding ItTen items in three containers force four into one of them. 367 people force a shared birthday. Neither statement tells you which container or which pair.ExplainerA Cone Is Exactly One Third of Its CylinderRadius 5, height 10: the cone holds 261.8 and the cylinder around it holds 785.4. The ratio is exactly a third, and it holds for any base shape at all.ExplainerOne Formula That Is Also a Rectangle and Also a TriangleParallel sides of 8 and 5 with a height of 4 give an area of 26 — the average width times the height. Make the two sides equal and it is a rectangle; shrink one to nothing and it is a triangle.ExplainerTwice as Wide Holds Four Times as Much; Twice as Tall Holds TwiceA cylinder of radius 5 and height 10 holds 785.4 units of volume. Double the radius and it holds 3,141.6; double the height instead and it holds 1,570.8.ExplainerForty Out and Sixty Back Averages Forty-EightThe harmonic mean of 40 and 60 is 48, not 50, because the slower leg takes longer and therefore counts for more. For 1, 2 and 4 it is 1.714 against an arithmetic mean of 2.33.ExplainerIf the Variance Is Not the Mean, It Is Not PoissonWith an average of five events per interval, exactly three occur 14.0 % of the time, none at all 0.7 %, and ten or more 3.2 % — about once every 31 intervals. The distribution has one parameter, and it sets both the mean and the variance.