In calculus, specifically when dealing with related rates, we often need to determine how the volume of an object changes over time (denoted as dV/dt) based on how its dimensions are changing. This concept is fundamental in physics, engineering, and fluid dynamics.
Common Mathematical Formulas
The rate of change is found by taking the derivative of the volume formula with respect to time (t) using the chain rule.
Sphere: If $V = \frac{4}{3}\pi r^3$, then $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.
Cube: If $V = s^3$, then $\frac{dV}{dt} = 3s^2 \frac{ds}{dt}$.
Cylinder: If $V = \pi r^2 h$ and the radius is constant, then $\frac{dV}{dt} = \pi r^2 \frac{dh}{dt}$.
Real-World Example
Imagine a spherical balloon being inflated. If the radius is currently 10 cm and is increasing at a rate of 2 cm/s, how fast is the volume increasing?
The volume is increasing at approximately 2,513.27 cm³/s.
Why This Matters
Calculating the rate of volume change is critical for determining how fast a tank fills, how quickly a glacier melts, or the rate at which pressure changes in a combustion chamber. Using this calculator, you can instantly find the derivative of volume for common shapes without manually performing the differentiation.
function updateInputs() {
var shape = document.getElementById("shapeSelect").value;
document.getElementById("sphereInputs").style.display = (shape === "sphere") ? "block" : "none";
document.getElementById("cubeInputs").style.display = (shape === "cube") ? "block" : "none";
document.getElementById("cylinderInputs").style.display = (shape === "cylinder") ? "block" : "none";
document.getElementById("resultArea").style.display = "none";
}
function calculateVolumeRate() {
var shape = document.getElementById("shapeSelect").value;
var dVdt = 0;
var formulaText = "";
var isValid = true;
if (shape === "sphere") {
var r = parseFloat(document.getElementById("radiusInput").value);
var drdt = parseFloat(document.getElementById("drdtInput").value);
if (isNaN(r) || isNaN(drdt)) {
isValid = false;
} else {
dVdt = 4 * Math.PI * Math.pow(r, 2) * drdt;
formulaText = "Formula: dV/dt = 4πr²(dr/dt)";
}
} else if (shape === "cube") {
var s = parseFloat(document.getElementById("sideInput").value);
var dsdt = parseFloat(document.getElementById("dsdtInput").value);
if (isNaN(s) || isNaN(dsdt)) {
isValid = false;
} else {
dVdt = 3 * Math.pow(s, 2) * dsdt;
formulaText = "Formula: dV/dt = 3s²(ds/dt)";
}
} else if (shape === "cylinder") {
var rCyl = parseFloat(document.getElementById("cylRadiusInput").value);
var dhdt = parseFloat(document.getElementById("dhdtInput").value);
if (isNaN(rCyl) || isNaN(dhdt)) {
isValid = false;
} else {
dVdt = Math.PI * Math.pow(rCyl, 2) * dhdt;
formulaText = "Formula: dV/dt = πr²(dh/dt)";
}
}
var resultArea = document.getElementById("resultArea");
var finalOutput = document.getElementById("finalOutput");
var calcFormula = document.getElementById("calcFormula");
if (!isValid) {
alert("Please enter valid numeric values for all fields.");
resultArea.style.display = "none";
return;
}
calcFormula.innerText = formulaText;
finalOutput.innerText = dVdt.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4});
resultArea.style.display = "block";
}