Used Formulas:
dV/dt = 4 × π × r² × (dr/dt)
dA/dt = 8 × π × r × (dr/dt)
function calculateSphereRates() {
// Get inputs
var radius = parseFloat(document.getElementById('rr_radius').value);
var drdt = parseFloat(document.getElementById('rr_drdt').value);
var unit = document.getElementById('rr_units').value;
var timeUnit = document.getElementById('rr_time_units').value;
// Validation
if (isNaN(radius) || isNaN(drdt)) {
alert("Please enter valid numeric values for Radius and Rate of Change.");
return;
}
if (radius < 0) {
alert("Radius cannot be negative in this physical context.");
return;
}
// Calculations
var PI = Math.PI;
// Volume Rate: dV/dt = 4 * pi * r^2 * dr/dt
var dVdt = 4 * PI * Math.pow(radius, 2) * drdt;
// Surface Area Rate: dA/dt = 8 * pi * r * dr/dt
var dAdt = 8 * PI * radius * drdt;
// Current Volume: V = (4/3) * pi * r^3
var currentVol = (4/3) * PI * Math.pow(radius, 3);
// Current Area: A = 4 * pi * r^2
var currentArea = 4 * PI * Math.pow(radius, 2);
// Formatting outputs
document.getElementById('res_dvdt').innerHTML = dVdt.toFixed(4) + " " + unit + "³/" + timeUnit;
document.getElementById('res_dadt').innerHTML = dAdt.toFixed(4) + " " + unit + "²/" + timeUnit;
document.getElementById('res_vol').innerHTML = currentVol.toFixed(4) + " " + unit + "³";
document.getElementById('res_area').innerHTML = currentArea.toFixed(4) + " " + unit + "²";
// Show results
document.getElementById('rr_results').style.display = 'block';
}
Understanding Related Rates in Spheres
In calculus, "related rates" problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The sphere is one of the most common geometric shapes used in these problems, often visualized as an inflating balloon, a melting snowball, or a growing tumor.
The Geometry of a Sphere
To solve related rates problems for a sphere, you must understand the relationship between its three primary variables: Radius (r), Surface Area (A), and Volume (V).
Volume Formula:V = (4/3)πr³
Surface Area Formula:A = 4πr²
Deriving the Rates (Differentiation)
When these variables change over time (t), we differentiate both sides of the geometry equations with respect to t using the chain rule.
1. Rate of Change of Volume (dV/dt)
Differentiating the volume formula yields:
dV/dt = 4πr² (dr/dt)
This equation tells us that the rate at which the volume increases depends on the current radius squared and the speed at which the radius is expanding.
2. Rate of Change of Surface Area (dA/dt)
Differentiating the surface area formula yields:
dA/dt = 8πr (dr/dt)
Example Problem: The Inflating Balloon
Imagine a spherical balloon is being inflated so that its radius increases at a rate of 2 cm/sec. How fast is the volume increasing when the radius is 10 cm?
Using the Calculator:
Enter 10 into the "Current Radius" field.
Enter 2 into the "Rate of Change of Radius" field.
Select "cm" and "sec" for units.
Click Calculate.
The Math Behind It: dV/dt = 4π(10)²(2) dV/dt = 4π(100)(2) dV/dt = 800π ≈ 2513.27 cm³/sec
Why is dr/dt Important?
The term dr/dt represents the velocity at which the boundary of the sphere is moving outwards. If dr/dt is negative, the sphere is shrinking (like a melting snowball), and consequently, dV/dt and dA/dt will also be negative.