Calculate the rate of change of one quantity with respect to time, given the rate of change of another related quantity.
Enter values to see the result.
Understanding Related Rates
Related rates problems are a common application of differential calculus. They involve finding the rate at which a quantity changes when we know the rate at which another related quantity changes, and the relationship between them.
The Core Idea: Implicit Differentiation
The fundamental technique used to solve related rates problems is implicit differentiation. Here's the process:
Identify Variables: Determine all the quantities that are changing and label them with distinct variables (e.g., r for radius, h for height, V for volume).
Find a Relationship: Establish an equation that connects these variables. This equation often comes from geometric formulas (like the area of a circle, volume of a sphere) or other physical principles.
Differentiate with Respect to Time: Differentiate both sides of the relationship equation with respect to time (t). Remember to use the chain rule for each variable that depends on time. For example, if A = πr², differentiating with respect to t gives dA/dt = 2πr (dr/dt).
Substitute Known Values: Plug in any given constant values and the rates of change (dr/dt, dA/dt, etc.) at the specific moment in time you are interested in.
Solve for the Unknown Rate: Solve the resulting equation for the unknown rate of change.
Example Calculation: Area of a Circle
Let's consider a classic example: The radius of a circle is increasing at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 5 cm?
Variables: Radius r, Area A.
Relationship: The formula for the area of a circle is A = πr².
Differentiate w.r.t. time (t): Using implicit differentiation and the chain rule:
d/dt (A) = d/dt (πr²) dA/dt = π * 2r * (dr/dt) dA/dt = 2πr (dr/dt)
Substitute Known Values: We are given: dr/dt = 2 cm/s.
We want to find dA/dt when r = 5 cm.
dA/dt = 2π(5 cm) * (2 cm/s)
Solve: dA/dt = 20π cm²/s
So, when the radius is 5 cm, the area is increasing at a rate of 20π square centimeters per second.
Calculator Usage
This calculator helps you perform these steps more quickly. You need to input:
The name of your first variable (e.g., 'Radius') and its rate of change (e.g., '2' if it's increasing at 2 units/sec).
The relationship between variables, expressed as a formula where the second variable is a function of the first (e.g., 'π * Radius^2' for Area if the first variable is 'Radius'). For powers, use '^' (e.g., 'x^3').
The rate of change for the second variable if it's known (leave blank or enter 0 if you are solving for it).
The specific value of the *first* variable at the moment of interest.
The calculator will attempt to evaluate the derivative symbolically and then substitute the values to find the unknown rate.
function calculateRelatedRate() {
var variable1Name = "r"; // Default to 'r' for simplicity in JS, actual naming is conceptual
var rate1Name = "dr/dt";
var variable2Name = "A"; // Default to 'A'
var rate2Name = "dA/dt";
var inputVariable1 = document.getElementById("variable1").value; // e.g. "Radius" – used for conceptual labeling
var inputRate1 = parseFloat(document.getElementById("rate1").value); // e.g. 2
var inputVariable2Formula = document.getElementById("variable2").value; // e.g. "π*r^2" or "A = π*r^2"
var inputRate2 = parseFloat(document.getElementById("rate2").value); // e.g. 0 or NaN
var inputSpecificValue = parseFloat(document.getElementById("specificValue").value); // e.g. 5
var resultDiv = document.getElementById("result");
resultDiv.style.backgroundColor = "#f8f9fa"; // Reset background
resultDiv.style.color = "var(–dark-text)"; // Reset text color
// Basic validation
if (isNaN(inputRate1) || isNaN(inputSpecificValue) || inputVariable2Formula.trim() === "") {
resultDiv.innerText = "Please enter valid numbers for rates and a formula.";
return;
}
// — Simplified Symbolic Differentiation and Evaluation —
// This is a highly simplified approach. Real symbolic differentiation is complex.
// We'll handle common cases like powers and simple constants/pi.
var derivedFormula = "";
var knownRateName = "";
var unknownRateName = "";
var calculatedRate = NaN;
var units = "units/sec"; // Placeholder units
// Attempt to parse the formula and find derivatives
try {
// Clean up formula input (remove ' = …', 'Variable Name = ')
if (inputVariable2Formula.includes('=')) {
inputVariable2Formula = inputVariable2Formula.split('=')[1].trim();
}
// Use a placeholder 'x' for the main variable in JS calculation
var formulaForEval = inputVariable2Formula.replace(/π/g, Math.PI.toString());
// — VERY Basic Symbolic Differentiation —
// This part is extremely limited and prone to errors for complex formulas.
// We focus on simple polynomial terms involving 'r' (our placeholder for variable1).
var diffExpression = "";
var parts = formulaForEval.split(/([\+\-])/); // Split by + or –
for (var i = 0; i 0 && diffExpression[diffExpression.length – 1] !== '(' && diffExpression[diffExpression.length-1] !== '+' && diffExpression[diffExpression.length-1] !== '-') ? " + " + newCoeff : newCoeff;
} else if (newExponent === 1) {
diffExpression += (diffExpression.length > 0 && diffExpression[diffExpression.length – 1] !== '(' && diffExpression[diffExpression.length-1] !== '+' && diffExpression[diffExpression.length-1] !== '-') ? " + " + newCoeff + "*r" : newCoeff + "*r";
} else {
diffExpression += (diffExpression.length > 0 && diffExpression[diffExpression.length – 1] !== '(' && diffExpression[diffExpression.length-1] !== '+' && diffExpression[diffExpression.length-1] !== '-') ? " + " + newCoeff + "*r^" + newExponent : newCoeff + "*r^" + newExponent;
}
} else {
// Cannot differentiate complex terms like (x+y)^2 or other variables
throw new Error("Unsupported formula complexity.");
}
} else if (term === 'r') { // Simple 'r' term (r^1)
// Derivative: coeff * 1 * r^(1-1) = coeff * r^0 = coeff
var newCoeff = coeff;
if (newCoeff !== 0) {
diffExpression += (diffExpression.length > 0 && diffExpression[diffExpression.length – 1] !== '(' && diffExpression[diffExpression.length-1] !== '+' && diffExpression[diffExpression.length-1] !== '-') ? " + " + newCoeff : newCoeff;
}
} else if (term.match(/^-?\d+(\.\d+)?$/) || term === 'π' || term === Math.PI.toString()) {
// If the term was just a number or Pi and had a coefficient, the derivative is 0.
// This case is handled by the initial coefficient check.
} else {
// Cannot differentiate terms not involving 'r'
throw new Error("Unsupported formula complexity.");
}
}
// Simplify the diffExpression a bit (e.g., remove leading '+ ')
if (diffExpression.startsWith('+ ')) {
diffExpression = diffExpression.substring(2);
}
// Handle case where derivative is 0
if (diffExpression === "") {
diffExpression = "0";
}
derivedFormula = diffExpression; // This is now the expression for dA/dt in terms of r and dr/dt
// — Evaluate the derived formula —
var rValue = inputSpecificValue;
var drdtValue = inputRate1;
var knownRateValue = inputRate2;
// Check if rate2 was provided (we are solving for rate1)
if (!isNaN(knownRateValue) && knownRateValue !== 0) {
unknownRateName = rate1Name; // Solving for dr/dt
// Formula is dA/dt = 2πr (dr/dt) => dr/dt = (dA/dt) / (2πr)
var denominator = 2 * Math.PI * rValue;
if (denominator === 0) {
resultDiv.innerText = "Error: Division by zero at the specific value.";
return;
}
calculatedRate = knownRateValue / denominator;
knownRateName = rate2Name;
units = "units/sec"; // Units for dr/dt
// Update labels conceptually
variable1Name = "Radius"; // Use the actual input name if available/needed later
variable2Name = "Area";
rate1Name = "Rate of Change of Radius";
rate2Name = "Rate of Change of Area";
} else { // Default: solving for rate2 (dA/dt)
unknownRateName = rate2Name; // Solving for dA/dt
knownRateName = rate1Name;
// Substitute r and dr/dt into the derived formula
var evalString = derivedFormula.replace(/r/g, rValue.toString());
// Handle potential multiple π's if user typed them
evalString = evalString.replace(/π/g, Math.PI.toString());
try {
calculatedRate = eval(evalString);
} catch (e) {
resultDiv.innerText = "Error evaluating formula. Check syntax.";
return;
}
units = "units²/sec"; // Units for dA/dt
// Update labels conceptually
variable1Name = "Radius"; // Use the actual input name if available/needed later
variable2Name = "Area";
rate1Name = "Rate of Change of Radius";
rate2Name = "Rate of Change of Area";
}
if (isNaN(calculatedRate)) {
resultDiv.innerText = "Calculation Error.";
} else {
resultDiv.style.backgroundColor = "var(–success-green)";
resultDiv.style.color = "white";
// Use the actual input names for context if they were provided simply
var displayVar1Name = inputVariable1.trim() ? inputVariable1.trim() : "Variable 1";
var displayVar2Name = inputVariable2Formula.split('=')[0].trim() ? inputVariable2Formula.split('=')[0].trim() : "Variable 2";
resultDiv.innerText = `${unknownRateName} = ${calculatedRate.toFixed(4)} ${units}`;
}
} catch (error) {
resultDiv.innerText = `Error: ${error.message}`;
}
}