Calculate continuous growth rates ($r$) based on the exponential model.
Relative Growth Rate ($r$):
Percentage Growth Rate:
Doubling Time ($T_d$):
Nature of Change:
function calculateGrowthRate() {
// 1. Get Elements
var initialInput = document.getElementById('initialValue');
var finalInput = document.getElementById('finalValue');
var timeInput = document.getElementById('timeElapsed');
var resultsArea = document.getElementById('resultsArea');
var errorDisplay = document.getElementById('errorDisplay');
var rateDecimalDisplay = document.getElementById('rateDecimal');
var ratePercentDisplay = document.getElementById('ratePercent');
var doublingTimeDisplay = document.getElementById('doublingTime');
var growthTypeDisplay = document.getElementById('growthType');
// 2. Parse Values
var N0 = parseFloat(initialInput.value);
var Nt = parseFloat(finalInput.value);
var t = parseFloat(timeInput.value);
// 3. Reset UI
errorDisplay.style.display = 'none';
resultsArea.style.display = 'none';
errorDisplay.innerHTML = ";
// 4. Validation
if (isNaN(N0) || isNaN(Nt) || isNaN(t)) {
errorDisplay.innerHTML = "Please enter valid numbers for all fields.";
errorDisplay.style.display = 'block';
return;
}
if (t === 0) {
errorDisplay.innerHTML = "Time elapsed cannot be zero.";
errorDisplay.style.display = 'block';
return;
}
if (N0 <= 0 || Nt 0) {
var dt = Math.log(2) / r;
timeMetric = dt.toFixed(2) + " units of time";
timeMetricLabel = "Doubling Time";
growthTypeDisplay.innerHTML = "Exponential Growth";
growthTypeDisplay.style.color = "green";
} else if (r < 0) {
var ht = Math.log(0.5) / r;
timeMetric = ht.toFixed(2) + " units of time";
timeMetricLabel = "Halving Time";
growthTypeDisplay.innerHTML = "Exponential Decay";
growthTypeDisplay.style.color = "red";
} else {
timeMetric = "Infinite";
timeMetricLabel = "Doubling Time";
growthTypeDisplay.innerHTML = "Stagnant (No Change)";
growthTypeDisplay.style.color = "gray";
}
// 6. Output Results
rateDecimalDisplay.innerHTML = r.toFixed(6);
ratePercentDisplay.innerHTML = rPercent.toFixed(4) + "%";
// Update the label for doubling/halving time dynamically
doublingTimeDisplay.parentElement.querySelector('.result-label').innerHTML = timeMetricLabel + ":";
doublingTimeDisplay.innerHTML = timeMetric;
resultsArea.style.display = 'block';
}
How to Calculate Relative Growth Rate Formula
Calculating the relative growth rate is a fundamental concept in fields ranging from biology (population dynamics) to finance (continuous compounding) and demographics. Unlike absolute growth, which simply tells you how much a quantity has increased, the relative growth rate tells you how fast the quantity is growing relative to its current size.
Understanding the Relative Growth Rate Formula
The relative growth rate, often denoted as r, is derived from the exponential growth model. The basic differential equation describing this growth is:
dN/dt = rN
Where:
dN/dt is the rate of change of the population/quantity over time.
N is the size of the population/quantity.
r is the constant relative growth rate.
Solving this differential equation gives us the standard exponential growth formula:
N(t) = N₀eʳᵗ
Solving for r (The Growth Rate)
To use this calculator, we rearrange the formula to solve specifically for the rate r based on an initial value, a final value, and the time elapsed:
r = ln(Nₜ / N₀) / t
Here is the breakdown of the variables:
ln = The natural logarithm (log base e).
Nₜ = The final value (population, mass, etc.) after time t.
N₀ = The initial value at the start.
t = The total time elapsed between the initial and final measurement.
Step-by-Step Calculation Example
Let's say you are studying a bacteria culture in a lab.
Identify Variables:
At 8:00 AM, the population is 100 cells (N₀).
At 12:00 PM (4 hours later), the population is 500 cells (Nₜ).
Apply Natural Log: Take the natural logarithm of the ratio.
ln(5) ≈ 1.6094.
Divide by Time: Divide the result by the time elapsed.
1.6094 / 4 ≈ 0.40235.
Result: The relative growth rate is approximately 0.40235 (or 40.235% per hour).
Difference Between Discrete and Continuous Growth
It is important to distinguish between simple percentage change and relative growth rate:
Simple Percentage Change: Calculates growth based on discrete intervals. Formula: $(Final – Initial) / Initial$. In the example above, this would be $(500-100)/100 = 400\%$. This assumes growth happened in one jump.
Relative Growth Rate (Continuous): Assumes growth is happening constantly at every instant. This is what the formula $r = \ln(N_t/N_0)/t$ calculates. In our example, the rate is roughly $40.2\%$.
Most scientific calculations regarding populations, radioactive decay, or continuous compound interest require the continuous relative growth rate formula used in the calculator above.