This tool calculates the constant growth rate required for an initial value to reach a final value over a specific period, assuming exponential growth (or decay). This is commonly used to determine Compound Annual Growth Rate (CAGR) in business, population growth in biology, or viral spread rates.
The unit of time determines the unit of the growth rate.
Understanding the Growth Rate Function
The exponential growth rate function measures how fast a quantity increases (or decreases) over time when the growth is proportional to the current value. Unlike linear growth, where a fixed amount is added each period, exponential growth compounds.
The Formula
The calculator determines the rate (r) using the rearranged exponential growth formula:
r = ((Vₜ / V₀)^(1/t)) – 1
V₀ (Initial Value): The starting amount at the beginning of the period.
Vₜ (Final Value): The ending amount after the time period has elapsed.
t (Time Period): The duration between the initial and final measurements.
r (Rate): The resulting growth rate per time unit (expressed as a decimal, converted to percentage in results).
Example Calculation
Suppose a bacterial culture starts with a population of 500. After 8 hours, the population has grown to 2,500. What is the hourly growth rate?
Initial Value (V₀) = 500
Final Value (Vₜ) = 2,500
Time Period (t) = 8
Calculation: r = ((2500 / 500)^(1/8)) – 1
r = (5^(0.125)) – 1 = 1.2228 – 1 = 0.2228
Converted to percentage: 22.28% growth per hour.
If the Final Value is less than the Initial Value, the result will be negative, indicating an exponential decay rate.
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function calculateGrowthRate() {
// 1. Get input values
var v0Str = document.getElementById('initialValue').value;
var vtStr = document.getElementById('finalValue').value;
var tStr = document.getElementById('timePeriod').value;
var resultBox = document.getElementById('growthResult');
// 2. Parse values to numbers
var v0 = parseFloat(v0Str);
var vt = parseFloat(vtStr);
var t = parseFloat(tStr);
// 3. Validate inputs
if (isNaN(v0) || isNaN(vt) || isNaN(t)) {
resultBox.style.display = "block";
resultBox.className = "result-box error-msg";
resultBox.innerHTML = "Please enter valid numeric values for all fields.";
return;
}
if (t === 0) {
resultBox.style.display = "block";
resultBox.className = "result-box error-msg";
resultBox.innerHTML = "Time period cannot be zero for this calculation.";
return;
}
if (v0 <= 0) {
resultBox.style.display = "block";
resultBox.className = "result-box error-msg";
resultBox.innerHTML = "Initial Value must be greater than zero for exponential growth calculation.";
return;
}
if (vt 0) {
descriptor = "Growth";
resultBox.className = "result-box"; // Default green
} else if (rateDecimal < 0) {
descriptor = "Decay/Decline";
resultBox.className = "result-box error-msg"; // Red for decline
} else {
descriptor = "No Change";
resultBox.className = "result-box";
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// 7. Display results
resultBox.style.display = "block";
resultBox.innerHTML = "Calculated " + descriptor + " Rate:" + ratePercent + "% per time unit";
}