Question

Two horizontal corridors, \(0 \leq x \leq a\) with \(y \geq 0\), and \(0 \leq y \leq b\) with \(x \geq 0\), meet at right angles. Find the length \(L\) of the longest ladder (considered as a stick) that may be carried horizontally around the corner.

Short answer

The length of the longest ladder is: \[ L = \text{√2} (a + b) \].

Step-by-step solution

Identify the Problem

We need to find the length of the longest ladder that can be carried horizontally around the corner formed by two corridors meeting at right angles. The corridors have dimensions: one extends from x = 0 to x = a with y ≥ 0, and the other extends from y = 0 to y = b with x ≥ 0.

Understand the Geometry

Visualize the corridors as two perpendicular lines meeting at the origin. We need to find the maximum length of a ladder that can pass around this right-angle corner when carried horizontally.

Set Up the Equation

Let the length of the ladder be L. As the ladder rotates around the corner, it can be split into two segments aligned to each corridor. At any instant, let the lengths of the ladder along each corridor be x and y such that: \[ x^2 + y^2 = L^2 \].

Define Constraints

The segments of the ladder along each corridor are restricted so that: x ≤ a and y ≤ b. Hence, we have the conditions: x ≤ a, y ≤ b, and \[ x^2 + y^2 = L^2 \].

Use Optimization

By maximizing the function constrained by the earlier conditions, we substitute x = a and y = b in the Pythagorean theorem: \[ L = \frac{a}{\text{cos}\theta} + \frac{b}{\text{sin}\theta} \]. By symmetry and calculus analysis, we find the maximum L.

Derive the Formula

The maximum L is given by: \[ L = \frac{a + b}{\text{sin}(\theta) + \text{cos}(\theta)} \]. Using θ = 45°, we get the maximum value. Substituting \[ \theta = \frac{\text{π}}{4} \], we get: \[ L = \frac{a + b}{\frac{2}{\text{√2}}} = \text{√2}(a + b) \].

Key concepts

geometry

Geometry is a branch of mathematics that deals with the properties and relations of points, lines, surfaces, and solids. In the context of the longest ladder problem, geometry helps us understand how the ladder fits within the confines of the perpendicular corridors.
Here’s what geometry teaches us about this situation:

• The corridors form a right-angle at the origin, intersecting at a point where one corridor spans horizontally and the other spans vertically.
• Visualizing the ladder as a line segment, we need to determine how it can span across this corner without violating the constraints set by the corridor walls.

By understanding the geometrical relationships, we can set up the problem correctly and find the appropriate mathematical solutions.

optimization

Optimization in mathematics involves finding the best solution from a set of possible choices. In the longest ladder problem, we aim to maximize the length of the ladder that can be carried horizontally around the corner.
This involves considering the following:

• A function representing the length of the ladder as it pivots around the corner,
• Constraints given by the dimensions of the corridors,

To find the optimal solution, we set up equations that represent these conditions and then use mathematical methods, such as calculus, to determine the maximum possible length of the ladder. This process ensures we find the longest possible ladder that fits the constraints.

Pythagorean theorem

The Pythagorean theorem is fundamental in solving the longest ladder problem. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Symbolically, it is expressed as: The Pythagorean theorem helps us relate the components of the ladder and corridors:

• Your ladder, when moving around the corner, splits into two right-angle triangle segments along the x and y axes.
• Let x and y be the lengths of the ladder segments along the corridors. According to the theorem, these are related by: \[ x^2 + y^2 = L^2 \]

This equation forms the basis for further calculations to determine the maximum length L of the ladder.

right-angle triangles

Right-angle triangles are key to understanding the longest ladder problem. A right-angle triangle has one 90-degree angle, and its sides can be related through the Pythagorean theorem.
The problem uses these properties in the following way:

• The corridors form a right-angle at their intersection.
• The ladder, as it navigates around the corner, forms a right-angle triangle with segments lying against each corridor.

Understanding this concept allows us to see how the ladder’s length is derived from its positioning in relation to the corridors.
This also simplifies the optimization process, making it easier to apply formulas and theorems to find the solution.