.calc-container {
font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
max-width: 800px;
margin: 20px auto;
padding: 25px;
border: 1px solid #ddd;
border-radius: 12px;
background-color: #f9f9f9;
color: #333;
box-shadow: 0 4px 6px rgba(0,0,0,0.1);
}
.calc-header {
text-align: center;
margin-bottom: 25px;
}
.calc-group {
margin-bottom: 15px;
}
.calc-label {
display: block;
font-weight: 600;
margin-bottom: 5px;
color: #2c3e50;
}
.calc-input, .calc-select {
width: 100%;
padding: 12px;
border: 1px solid #ccc;
border-radius: 6px;
box-sizing: border-box;
font-size: 16px;
}
.calc-btn {
width: 100%;
background-color: #3498db;
color: white;
padding: 15px;
border: none;
border-radius: 6px;
font-size: 18px;
font-weight: bold;
cursor: pointer;
transition: background-color 0.3s;
}
.calc-btn:hover {
background-color: #2980b9;
}
.calc-result {
margin-top: 25px;
padding: 20px;
background-color: #e8f4fd;
border-left: 5px solid #3498db;
border-radius: 4px;
display: none;
}
.calc-result h3 {
margin-top: 0;
color: #2c3e50;
}
.article-section {
margin-top: 40px;
line-height: 1.6;
}
.article-section h2 {
color: #2c3e50;
border-bottom: 2px solid #3498db;
padding-bottom: 10px;
}
.example-box {
background: #fff;
border: 1px dashed #3498db;
padding: 15px;
margin: 15px 0;
}
Select Geometric Scenario
Circle (Area relative to Radius)
Sphere (Volume relative to Radius)
Sphere (Surface Area relative to Radius)
Square (Area relative to Side)
Cube (Volume relative to Side)
Current Radius (r) or Side (s)
Rate of Change of Radius/Side (dr/dt or ds/dt)
Calculate Derivative Rate
Calculated Rate of Change
Understanding Related Rates in Calculus
In differential calculus, related rates problems involve finding the rate at which a certain quantity changes by relating that quantity to other quantities whose rates of change are already known. The process typically involves the chain rule to differentiate both sides of an equation with respect to time (t).
The Core Formula
When we say two variables are related, we mean there is an equation connecting them. For example, the area of a circle is linked to its radius by $A = \pi r^2$. If the radius is changing over time ($dr/dt$), the area must also be changing over time ($dA/dt$).
Using the chain rule: dA/dt = 2πr * (dr/dt)
Realistic Example: The Expanding Balloon
Imagine a spherical balloon being inflated. If the radius is currently 10 cm and the radius is increasing at a rate of 0.5 cm/s , how fast is the volume increasing?
Knowns: r = 10, dr/dt = 0.5Formula: V = (4/3)πr³Derivative: dV/dt = 4πr² * (dr/dt)Calculation: dV/dt = 4 * π * (10)² * 0.5 = 200π ≈ 628.32 cm³/s
Common Related Rate Formulas
Circle Area: dA/dt = 2πr(dr/dt)Sphere Volume: dV/dt = 4πr²(dr/dt)Sphere Surface Area: dS/dt = 8πr(dr/dt)Square Area: dA/dt = 2s(ds/dt)Cube Volume: dV/dt = 3s²(ds/dt)
How to Use This Calculator
Select your shape: Choose the geometric relationship you are solving for.Input the current value: Enter the measurement of the radius or side at the specific moment in time.Input the rate: Enter how fast that radius or side is growing (positive) or shrinking (negative).Analyze the Result: The calculator provides the instantaneous rate of change for the primary property (Volume or Area).
function updateLabels() {
var shape = document.getElementById("shapeType").value;
var valLabel = document.getElementById("valLabel");
var rateLabel = document.getElementById("rateLabel");
if (shape === "circleArea" || shape === "sphereVolume" || shape === "sphereSurface") {
valLabel.innerText = "Current Radius (r)";
rateLabel.innerText = "Rate of Change of Radius (dr/dt)";
} else {
valLabel.innerText = "Current Side Length (s)";
rateLabel.innerText = "Rate of Change of Side (ds/dt)";
}
}
function calculateRelatedRate() {
var shape = document.getElementById("shapeType").value;
var val = parseFloat(document.getElementById("currentValue").value);
var rate = parseFloat(document.getElementById("currentRate").value);
var outputDiv = document.getElementById("resultArea");
var outputText = document.getElementById("calcOutput");
var formulaText = document.getElementById("formulaOutput");
if (isNaN(val) || isNaN(rate)) {
alert("Please enter valid numerical values for both inputs.");
return;
}
var result = 0;
var formulaUsed = "";
var unitLabel = "";
if (shape === "circleArea") {
// dA/dt = 2 * PI * r * dr/dt
result = 2 * Math.PI * val * rate;
formulaUsed = "dA/dt = 2πr(dr/dt)";
unitLabel = "units²/time";
} else if (shape === "sphereVolume") {
// dV/dt = 4 * PI * r^2 * dr/dt
result = 4 * Math.PI * Math.pow(val, 2) * rate;
formulaUsed = "dV/dt = 4πr²(dr/dt)";
unitLabel = "units³/time";
} else if (shape === "sphereSurface") {
// dS/dt = 8 * PI * r * dr/dt
result = 8 * Math.PI * val * rate;
formulaUsed = "dS/dt = 8πr(dr/dt)";
unitLabel = "units²/time";
} else if (shape === "squareArea") {
// dA/dt = 2 * s * ds/dt
result = 2 * val * rate;
formulaUsed = "dA/dt = 2s(ds/dt)";
unitLabel = "units²/time";
} else if (shape === "cubeVolume") {
// dV/dt = 3 * s^2 * ds/dt
result = 3 * Math.pow(val, 2) * rate;
formulaUsed = "dV/dt = 3s²(ds/dt)";
unitLabel = "units³/time";
}
outputText.innerHTML = "The instantaneous rate of change is
" + result.toFixed(4) + " " + unitLabel + ".";
formulaText.innerText = "Formula used: " + formulaUsed;
outputDiv.style.display = "block";
}