.calc-header {
text-align: center;
margin-bottom: 25px;
background: #f8f9fa;
padding: 20px;
border-radius: 8px;
border-bottom: 3px solid #007bff;
}
.calc-header h2 {
margin: 0;
color: #333;
font-size: 24px;
}
.calc-row {
display: flex;
flex-wrap: wrap;
gap: 20px;
margin-bottom: 20px;
}
.calc-col {
flex: 1;
min-width: 250px;
}
.form-group {
margin-bottom: 15px;
}
.form-group label {
display: block;
margin-bottom: 8px;
font-weight: 600;
color: #555;
font-size: 14px;
}
.form-group input {
width: 100%;
padding: 12px;
border: 1px solid #ddd;
border-radius: 6px;
font-size: 16px;
transition: border-color 0.2s;
box-sizing: border-box;
}
.form-group input:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 3px rgba(0,123,255,0.1);
}
.calc-btn {
width: 100%;
padding: 14px;
background-color: #007bff;
color: white;
border: none;
border-radius: 6px;
font-size: 16px;
font-weight: 700;
cursor: pointer;
transition: background-color 0.2s;
text-transform: uppercase;
letter-spacing: 0.5px;
}
.calc-btn:hover {
background-color: #0056b3;
}
.results-box {
margin-top: 25px;
background: #f0f7ff;
border: 1px solid #cce5ff;
border-radius: 8px;
padding: 20px;
display: none;
}
.result-row {
display: flex;
justify-content: space-between;
align-items: center;
padding: 10px 0;
border-bottom: 1px solid #dae0e5;
}
.result-row:last-child {
border-bottom: none;
}
.result-label {
color: #495057;
font-weight: 500;
}
.result-value {
font-weight: 700;
color: #007bff;
font-size: 18px;
}
.highlight-value {
font-size: 24px;
color: #28a745;
}
.error-msg {
color: #dc3545;
text-align: center;
margin-top: 10px;
font-weight: 600;
display: none;
}
.article-section {
margin-top: 40px;
line-height: 1.6;
color: #333;
}
.article-section h2 {
font-size: 22px;
margin-top: 30px;
margin-bottom: 15px;
color: #2c3e50;
border-left: 4px solid #007bff;
padding-left: 10px;
}
.article-section p {
margin-bottom: 15px;
}
.article-section ul {
margin-bottom: 15px;
padding-left: 20px;
}
.article-section li {
margin-bottom: 8px;
}
.formula-box {
background: #f8f9fa;
padding: 15px;
border-radius: 6px;
font-family: "Courier New", Courier, monospace;
text-align: center;
font-weight: bold;
border: 1px dashed #adb5bd;
margin: 20px 0;
}
function calculateContinuousRate() {
var initialVal = document.getElementById('initialValue').value;
var rateVal = document.getElementById('growthRate').value;
var timeVal = document.getElementById('timeElapsed').value;
var errorDiv = document.getElementById('error-message');
var resultsDiv = document.getElementById('results');
// Reset display
errorDiv.style.display = 'none';
resultsDiv.style.display = 'none';
// Validation
if (initialVal === " || rateVal === " || timeVal === ") {
errorDiv.innerText = "Please fill in all fields.";
errorDiv.style.display = 'block';
return;
}
var P = parseFloat(initialVal);
var r = parseFloat(rateVal); // Percentage
var t = parseFloat(timeVal);
if (isNaN(P) || isNaN(r) || isNaN(t)) {
errorDiv.innerText = "Please enter valid numeric values.";
errorDiv.style.display = 'block';
return;
}
// Calculation: P(t) = P * e^(r * t)
// Note: r needs to be converted from percentage to decimal (r/100)
var rateDecimal = r / 100;
var finalResult = P * Math.exp(rateDecimal * t);
var totalChange = finalResult – P;
var percentChange = ((finalResult – P) / P) * 100;
// Display Results
// We use toLocaleString for pretty number formatting
document.getElementById('finalValue').innerText = finalResult.toLocaleString(undefined, {
minimumFractionDigits: 2,
maximumFractionDigits: 4
});
var changeSign = totalChange >= 0 ? "+" : "";
document.getElementById('totalChange').innerText = changeSign + totalChange.toLocaleString(undefined, {
minimumFractionDigits: 2,
maximumFractionDigits: 4
});
document.getElementById('pctChange').innerText = changeSign + percentChange.toLocaleString(undefined, {
minimumFractionDigits: 2,
maximumFractionDigits: 2
}) + "%";
resultsDiv.style.display = 'block';
}
Continuous Rate Calculator
Calculate exponential growth or decay using the continuous compounding formula.
Starting population, principal, or mass
Percentage % (use negative for decay)
Years, hours, or relevant time unit
Please enter valid numeric values for all fields.
Final Value ($P(t)$):
0
Total Change:
0
Percentage Growth/Decay:
0%
Formula Used:
P(t) = P₀ · e^(rt)
Understanding the Continuous Rate Calculator
The Continuous Rate Calculator is a powerful mathematical tool designed to compute the future value of a quantity undergoing continuous growth or decay. Unlike discrete compounding (which calculates changes at set intervals like annually or monthly), continuous rate calculations assume that growth occurs at every possible instant of time. This concept is fundamental in fields ranging from physics and biology to finance.
The Continuous Growth Formula
Calculations are based on the standard exponential growth formula involving the mathematical constant e (Euler's number, approximately 2.71828).
P(t) = P₀ · e^(rt)
Where:
- P(t): The final value at time t.
- P₀: The initial value (Initial Amount).
- e: Euler's number (base of the natural logarithm).
- r: The continuous rate of growth or decay (entered as a percentage).
- t: The time period elapsed.
When to Use Continuous Rate?
This calculator is essential for scenarios where changes happen constantly rather than in steps:
- Population Growth: Modeling the growth of bacteria, cells, or human populations where birth and death are continuous processes.
- Radioactive Decay: Calculating the remaining mass of a radioactive isotope over time (using a negative rate).
- Thermodynamics: Newton's Law of Cooling often utilizes continuous rate principles.
- Continuous Compounding: In finance, determining the theoretical upper limit of compound interest where interest is added an infinite number of times per period.
Example Calculation
Suppose you are studying a bacterial culture. You start with an Initial Value of 100 bacteria. The culture grows at a Continuous Rate of 50% per hour. You want to know the population after 4 hours.
- P₀ = 100
- r = 50% (0.50)
- t = 4
Using the formula: 100 · e^(0.50 × 4) = 100 · e^2 ≈ 100 · 7.389 = 738.9.
The calculator performs this complex math instantly, giving you the precise ending value and the total growth percentage.