The "Ladder Sliding Down a Wall" is a classic problem in Calculus I that demonstrates the concept of related rates. It involves a ladder of fixed length leaning against a vertical wall. As the base of the ladder slides away from the wall, the top of the ladder must slide down the wall.
The Mathematical Foundation
This problem is solved using the Pythagorean Theorem, which relates the sides of a right triangle:
x² + y² = L²
Where:
x = Horizontal distance from the wall to the base of the ladder.
y = Vertical distance from the floor to the top of the ladder.
L = The constant length of the ladder.
The Derivative (The "Rates" Part)
Since the ladder is moving, x and y are functions of time (t). To find how fast the top is sliding down (dy/dt) based on how fast the base is moving (dx/dt), we take the derivative of the equation with respect to t:
2x(dx/dt) + 2y(dy/dt) = 0
Because the ladder length (L) is constant, its derivative is zero. Rearranging for the unknown rate (dy/dt) gives us:
dy/dt = – (x / y) * (dx/dt)
Example Calculation
Suppose you have a 13-foot ladder leaning against a wall. The base is 5 feet from the wall and is being pulled away at a rate of 2 feet per second. How fast is the top sliding down?
The negative sign indicates that the height y is decreasing (the ladder is sliding down).
Common Pitfalls
Units: Ensure all measurements (feet, meters, seconds) are consistent.
Signs: Remember that if the base is moving away from the wall, dx/dt is positive. If the top is moving down, dy/dt will be negative.
Constants: The length of the ladder (L) is almost always constant, so its derivative is 0.
function calculateLadderRates() {
var L = parseFloat(document.getElementById('ladderLength').value);
var x = parseFloat(document.getElementById('baseDistance').value);
var dxdt = parseFloat(document.getElementById('baseRate').value);
var resultDiv = document.getElementById('ladderResult');
var heightRes = document.getElementById('heightResult');
var rateRes = document.getElementById('rateResult');
var formulaRes = document.getElementById('formulaDisplay');
// Validation
if (isNaN(L) || isNaN(x) || isNaN(dxdt)) {
alert("Please enter valid numerical values for all fields.");
return;
}
if (x >= L) {
alert("The distance from the wall (x) cannot be greater than or equal to the ladder length (L).");
return;
}
if (L <= 0 || x < 0) {
alert("Length must be positive and distance cannot be negative.");
return;
}
// 1. Calculate y using Pythagorean theorem: x^2 + y^2 = L^2
var ySq = (L * L) – (x * x);
var y = Math.sqrt(ySq);
// 2. Calculate dy/dt using derivative: 2x(dx/dt) + 2y(dy/dt) = 0
// dy/dt = -(x * dx/dt) / y
var dydt = -(x * dxdt) / y;
// Display Results
resultDiv.style.display = "block";
heightRes.innerHTML = "Height of the top of the ladder (y): " + y.toFixed(3) + " units";
var direction = dydt < 0 ? "sliding down" : "sliding up";
rateRes.innerHTML = "Rate of change of height (dy/dt): " + dydt.toFixed(3) + " units/sec (" + direction + ")";
formulaRes.innerHTML = "Step 1: y = √(" + L + "² – " + x + "²) = " + y.toFixed(3) + "" +
"Step 2: dy/dt = -(" + x + " / " + y.toFixed(3) + ") * " + dxdt + " = " + dydt.toFixed(3);
}