Question

A steel girder \(25 \mathrm{ft}\). long is moved on rollers along a passageway \(12.8 \mathrm{ft}\). wide, and into a corridor at right angles to the passageway. Neglecting the horizontal width of the girder, find how wide the corridor must be in order that the girder may go round the corner.

Short answer

Answer: The minimum width of the corridor should be approximately 21.47 ft.

Step-by-step solution

Draw a diagram to visualize the problem

To help solve the problem, draw a right-angled triangle with the steel girder as the hypotenuse, the passageway width as one side, and the corridor width as the other side.

Apply the Pythagorean theorem

Use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where a and b are the legs of the right-angled triangle (the widths of the passageway and corridor), and c is the hypotenuse (the length of the steel girder). In this case, a = 12.8 ft (passageway width), and c = 25 ft (steel girder length). We want to find the width of the corridor (b).

Rearrange the equation to solve for the corridor width (b)

Rearrange the Pythagorean theorem to solve for b: \(b^2 = c^2 - a^2\) Plug in the known values: \(b^2 = (25 \mathrm{ft})^2 - (12.8 \mathrm{ft})^2\)

Calculate the corridor width (b)

Calculate the corridor width: \(b^2 = 625 \mathrm{ft}^2 - 163.84 \mathrm{ft}^2\) \(b^2 = 461.16 \mathrm{ft}^2\) \(b = \sqrt{461.16 \mathrm{ft}^2}\) \(b \approx 21.47 \mathrm{ft}\)

State the conclusion

The corridor must be at least approximately \(21.47 \mathrm{ft}\) wide for the steel girder to go around the corner.

Key concepts

Geometry and Its Role in Problem Solving

Geometry is the branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Among its primary uses is to solve practical problems, such as determining the space required to navigate objects through various environments.

In the exercise involving the steel girder, we use geometrical concepts to visualize and solve a real-world problem. The process begins with visual representation, often by drawing geometrical shapes that relate to the problem at hand. In this scenario, a right-angled triangle is formed by the girder's path through the passageway and into the corridor, providing a framework for applying geometrical theorems.

Geometrical visualization turns abstract problems into tangible representations, making it easier to understand and work through complex spatial issues. This approach leverages basic shapes and the properties of these shapes, such as the fixed relationship between the sides of right-angled triangles captured by the Pythagorean theorem.

Understanding Right-Angled Triangles

Right-angled triangles are a fundamental element of geometry, characterized by one angle measuring exactly 90 degrees. This unique property leads to several theorems that are exclusive to these triangles, the most renowned being the Pythagorean theorem.

The significance of right-angled triangles extends beyond their theoretical properties. They are a practical tool used in a variety of fields, including architecture, engineering, and even navigation. For instance, the given problem involving the steel girder illustrates how the knowledge of the right-angled triangle is instrumental in solving issues related to spatial movement.

Moreover, right-angled triangles provide an excellent starting point for students learning trigonometry, as they offer clear examples for the sine, cosine, and tangent functions—which are based on the ratios of a right-angled triangle's sides.

The Art of Math Problem Solving

Math problem solving is not just about crunching numbers; it's a strategic and analytical process that requires understanding the relationship between the mathematical concepts and the context of the problem. Here's how problem-solving is demonstrated in the exercise:

Visualizing the Problem

Diagrams are crucial in grasping the problem's spatial elements, laying a clear path toward the solution.

Applying Mathematical Theorems

In our exercise, the Pythagorean theorem is the key to unlocking the relationship between the lengths of the triangle's sides. Math problem solving often involves identifying and applying the right theorem or formula.

Manipulating Algebraic Expressions

To isolate the desired variable, in this case, the width of the corridor, algebraic rearrangement is used, demonstrating the importance of algebra in problem-solving.

The approach taken to solve the girder problem highlights these problem-solving stages. Each step serves a specific purpose in guiding us from initial confusion to arriving at a logical and practical solution.