Continuous growth is a fundamental concept in mathematics and science used to describe processes where change happens at every single instant, rather than at discrete intervals. Unlike simple or standard compound growth, which adds value at specific steps (like annually or monthly), continuous growth implies that the compounding period is infinitesimally small.
This calculator is versatile and can solve for any variable in the continuous growth formula. It is commonly used for:
Population Dynamics: Estimating the future population of a city or bacterial colony.
Physics: Modeling radioactive decay (using a negative rate).
Biology: Tracking the spread of a virus or cell division.
The Continuous Growth Formula
The mathematical model used for continuous growth (or exponential growth) is the natural exponential function:
A = P · ert
Where:
A = The Final Amount (Ending Value) after time t.
P = The Initial Amount (Principal or Starting Value).
e = Euler's Number (approximately 2.71828). This is a mathematical constant that is the base of natural logarithms.
r = The Rate of growth (or decay) per time period (expressed as a decimal).
t = The Time duration.
Variations of the Formula
Depending on what information you have, you can rearrange the formula to solve for the missing variable:
To find Initial Value (P): P = A / ert
To find Rate (r): r = ln(A / P) / t
To find Time (t): t = ln(A / P) / r
Note: "ln" stands for the natural logarithm, which is the inverse function of the exponential function e.
Example Calculation
Let's say you are studying a bacterial culture. You start with 100 bacteria (P). The bacteria grow at a continuous rate of 20% per hour (r = 0.20). You want to know how many bacteria there will be after 5 hours (t).
Using the formula A = Pert:
A = 100 · e(0.20 × 5)
A = 100 · e1
A = 100 · 2.71828…
A ≈ 271.83
After 5 hours, you would have approximately 272 bacteria.
Continuous Decay
This same calculator handles decay. If you are calculating radioactive decay or population decline, simply enter a negative number for the rate. For example, a rate of -5% means the quantity is decreasing continuously by 5%.
Continuous vs. Discrete Compounding
The difference between standard compound growth (like a bank account compounding monthly) and continuous growth is the frequency of compounding.
Discrete: Interest is added at set intervals (e.g., 12 times a year).
Continuous: Interest is added at every possible instant.
Continuous growth always yields a slightly higher result than discrete compounding for the same rate and time, representing the theoretical upper limit of growth.
function toggleInputs() {
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// Reset visibility
document.getElementById('group-P').style.display = 'block';
document.getElementById('group-A').style.display = 'block';
document.getElementById('group-r').style.display = 'block';
document.getElementById('group-t').style.display = 'block';
// Hide the input we are calculating for
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function calculateGrowth() {
// 1. Determine Mode
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var mode = 'A';
for (var i = 0; i < radios.length; i++) {
if (radios[i].checked) {
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// 2. Get Input Values
var P = parseFloat(document.getElementById('initialValue').value);
var A = parseFloat(document.getElementById('finalValue').value);
var r_percent = parseFloat(document.getElementById('rate').value);
var t = parseFloat(document.getElementById('time').value);
// Convert percentage to decimal
var r = r_percent / 100;
var result = 0;
var formulaText = "";
var resultLabelText = "";
// 3. Validation & Calculation
if (mode === 'A') {
// Solving for Final Amount
if (isNaN(P) || isNaN(r_percent) || isNaN(t)) {
alert("Please enter valid numbers for Initial Value, Rate, and Time.");
return;
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result = P * Math.exp(r * t);
formulaText = "Formula Used: A = P · e^(rt)";
resultLabelText = "Final Amount (A)";
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else if (mode === 'P') {
// Solving for Initial Amount
if (isNaN(A) || isNaN(r_percent) || isNaN(t)) {
alert("Please enter valid numbers for Final Value, Rate, and Time.");
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result = A / Math.exp(r * t);
formulaText = "Formula Used: P = A / e^(rt)";
resultLabelText = "Initial Value (P)";
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else if (mode === 'r') {
// Solving for Rate
if (isNaN(P) || isNaN(A) || isNaN(t)) {
alert("Please enter valid numbers for Initial Value, Final Value, and Time.");
return;
}
if (P <= 0 || A <= 0) {
alert("Initial and Final Values must be positive to calculate rate.");
return;
}
// r = ln(A/P) / t
var rateDecimal = Math.log(A / P) / t;
result = rateDecimal * 100; // Convert back to percent
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// Solving for Time
if (isNaN(P) || isNaN(A) || isNaN(r_percent)) {
alert("Please enter valid numbers for Initial Value, Final Value, and Rate.");
return;
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if (P <= 0 || A <= 0) {
alert("Initial and Final Values must be positive to calculate time.");
return;
}
if (r === 0) {
alert("Rate cannot be zero for this calculation.");
return;
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// t = ln(A/P) / r
result = Math.log(A / P) / r;
formulaText = "Formula Used: t = ln(A/P) / r";
resultLabelText = "Time Period (t)";
}
// 4. Formatting Result
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// Format based on what we are calculating
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} else if (mode === 't') {
formattedResult = result.toFixed(2) + " time units";
} else {
// For amounts, use commas but no currency symbol (generic growth)
formattedResult = result.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 2});
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