In calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time (t).
These problems are solved using Implicit Differentiation. Because the variables are functions of time, we apply the Chain Rule when differentiating. For example, if you are differentiating volume V with respect to time, the derivative is dV/dt.
Common Scenarios Explained
1. The Expanding Sphere
Imagine a balloon being inflated. As the radius grows, the volume grows much faster.
Formula: V = (4/3)πr³
Differentiation: dV/dt = 4πr²(dr/dt)
This calculator solves for the rate of volume change (dV/dt) given the current radius and how fast the radius is expanding.
2. The Sliding Ladder
A classic physics and calculus problem. A ladder leans against a wall. If the bottom slides away, how fast does the top slide down?
Formula: x² + y² = L² (Pythagorean Theorem)
Differentiation: 2x(dx/dt) + 2y(dy/dt) = 0
Result: dy/dt = – (x / y) * (dx/dt)
Note: The result is usually negative, indicating the top of the ladder is moving downwards.
3. The Filling Cone
Water is poured into a conical tank. The radius of the water surface and the depth of the water are related by similar triangles based on the tank's dimensions.
Formula: V = (1/3)πr²h
Constraint: r = h(R/H)
Substituted Formula: V = (π/3)(R/H)²h³
Differentiation: dV/dt = π(R/H)²h²(dh/dt)
How to Use This Calculator
Select Scenario: Choose the geometry that matches your problem (Sphere, Circle, Ladder, or Cone).
Enter Knowns: Input the static dimensions (like Ladder Length) and the instantaneous values (Current Radius, Current Distance).
Enter Rates: Input the known rate of change (e.g., how fast the radius is growing).
Calculate: Click the button to find the unknown related rate.
function toggleInputs() {
var type = document.getElementById('problemType').value;
var sections = document.getElementsByClassName('calc-section');
for (var i = 0; i = L) throw "Base distance cannot be greater than or equal to Ladder length.";
// Calculate y
var y = Math.sqrt(Math.pow(L, 2) – Math.pow(x, 2));
// dy/dt = – (x / y) * dx/dt
calculatedValue = -(x / y) * dxdt;
titleText = "Velocity of Ladder Top (dy/dt)";
formulaText = "y = " + y.toFixed(2) + "; dy/dt = -(" + x + "/" + y.toFixed(2) + ") * " + dxdt;
} else if (type === 'cone') {
var R = parseFloat(document.getElementById('coneTotalRadius').value);
var H = parseFloat(document.getElementById('coneTotalHeight').value);
var h = parseFloat(document.getElementById('coneWaterLevel').value);
var dhdt = parseFloat(document.getElementById('coneRate').value);
if (isNaN(R) || isNaN(H) || isNaN(h) || isNaN(dhdt)) throw "Please enter valid numbers.";
if (H === 0) throw "Height cannot be zero.";
// V = (1/3) * pi * (R/H * h)^2 * h = (pi/3) * (R/H)^2 * h^3
// dV/dt = pi * (R/H)^2 * h^2 * dh/dt
var ratio = R / H;
calculatedValue = Math.PI * Math.pow(ratio, 2) * Math.pow(h, 2) * dhdt;
titleText = "Rate of Water Volume Change (dV/dt)";
formulaText = "dV/dt = π(R/H)²h²(dh/dt)";
}
resultValue.innerHTML = calculatedValue.toFixed(4);
resultTitle.innerHTML = titleText;
resultFormula.innerHTML = formulaText;
resultArea.style.display = 'block';
} catch (err) {
errorDisplay.innerHTML = err;
errorDisplay.style.display = 'block';
}
}