Calculus Related Rates Calculator
Expanding Circle (Area)
Expanding Sphere (Volume)
Expanding Cube (Volume)
Sliding Ladder (Pythagorean)
Result: Rate of Area Change (dA/dt)
" + "Given Radius (r): " + r + "" + "Given Rate (dr/dt): " + drdt + "" + "Formula: dA/dt = 2πr(dr/dt)" + "dA/dt = " + resultValue.toFixed(4) + " units²/sec"; } else if (scenario === "sphere") { // V = 4/3 * pi * r^3 -> dV/dt = 4 * pi * r^2 * dr/dt var r = val1; var drdt = val2; resultValue = 4 * Math.PI * Math.pow(r, 2) * drdt; outputHTML = "Result: Rate of Volume Change (dV/dt)
" + "Given Radius (r): " + r + "" + "Given Rate (dr/dt): " + drdt + "" + "Formula: dV/dt = 4πr²(dr/dt)" + "dV/dt = " + resultValue.toFixed(4) + " units³/sec"; } else if (scenario === "cube") { // V = s^3 -> dV/dt = 3 * s^2 * ds/dt var s = val1; var dsdt = val2; resultValue = 3 * Math.pow(s, 2) * dsdt; outputHTML = "Result: Rate of Volume Change (dV/dt)
" + "Given Side (s): " + s + "" + "Given Rate (ds/dt): " + dsdt + "" + "Formula: dV/dt = 3s²(ds/dt)" + "dV/dt = " + resultValue.toFixed(4) + " units³/sec"; } else if (scenario === "ladder") { // x^2 + y^2 = L^2 // 2x(dx/dt) + 2y(dy/dt) = 0 -> dy/dt = -(x/y)(dx/dt) var x = val1; var dxdt = val2; var L = parseFloat(document.getElementById("input3").value); if (isNaN(L)) { resultDiv.style.display = "block"; resultDiv.innerHTML = "Please enter the Ladder Length."; return; } if (x >= L) { resultDiv.style.display = "block"; resultDiv.innerHTML = "Distance from wall (x) cannot be greater than or equal to Ladder Length (L)."; return; } var y = Math.sqrt(Math.pow(L, 2) – Math.pow(x, 2)); resultValue = -1 * (x / y) * dxdt; outputHTML = "Result: Vertical Velocity (dy/dt)
" + "Derived Height on Wall (y): " + y.toFixed(4) + "" + "Formula: dy/dt = -(x/y)(dx/dt)" + "dy/dt = " + resultValue.toFixed(4) + " units/sec" + "Note: A negative value indicates the top of the ladder is sliding down."; } resultDiv.innerHTML = outputHTML; resultDiv.style.display = "block"; }Understanding Related Rates in Calculus
Related rates problems are among the most common applications of differentiation in calculus. They involve finding a rate at which a quantity changes by relating it to other quantities whose rates of change are known. The "rate of change" is usually with respect to time ($t$).
The Core Concept: The Chain Rule
The fundamental tool for solving these problems is the Chain Rule. When we differentiate an equation with respect to time $t$, we are implicitly differentiating every variable. For example, if you have a variable $r$ (radius) that changes over time, its derivative is not just 1; it is $\frac{dr}{dt}$.
Common Scenarios Covered by This Calculator
- Expanding Circle: Often used to model ripples in a pond or oil spills. We relate the change in Area ($A$) to the change in Radius ($r$) using $A = \pi r^2$.
- Expanding Sphere: Useful for inflating balloons. We relate the change in Volume ($V$) to the change in Radius ($r$) using $V = \frac{4}{3}\pi r^3$.
- Expanding Cube: Relates Volume ($V$) to the side length ($s$) using $V = s^3$.
- Sliding Ladder: A classic Pythagorean theorem problem. A ladder leans against a wall; as the base slides out, the top slides down. We use $x^2 + y^2 = L^2$ to relate the velocities.
How to Solve Related Rates Problems Manually
If you are a student, it is important to know the steps to solve these without a calculator:
- Draw a Picture: Label all constant values (like the length of a ladder) and variable values (like distance $x$ or radius $r$).
- Identify Given and Required Rates: Write down what you know (e.g., $\frac{dx}{dt} = 2$) and what you need to find (e.g., $\frac{dy}{dt} = ?$).
- Write the Equation: Find a geometric formula that relates the variables (Area, Volume, Pythagorean Theorem, Trig ratios).
- Differentiate with Respect to Time ($t$): Apply the Chain Rule. Remember that the derivative of $x^2$ becomes $2x \frac{dx}{dt}$.
- Substitute and Solve: Plug in the known values for the specific instant in time and solve for the unknown rate.