Finding the Longest Pole in a Rectangular Room

The question asks for the length of the longest pole that can fit inside a rectangular room with given dimensions. A rectangular room is essentially a rectangular prism or a cuboid. The longest straight line that can be drawn within a cuboid extends from one corner to the opposite corner that is furthest away. This line is known as the space diagonal of the cuboid.

Understanding the Space Diagonal

For a rectangular prism with dimensions length ($l$), width ($w$), and height ($h$), the length of the space diagonal ($d$) can be calculated using the Pythagorean theorem extended to three dimensions. The formula for the space diagonal is:

\( d = \sqrt{l^2 + w^2 + h^2} \)

This formula essentially involves finding the diagonal of the base rectangle first (using Pythagorean theorem in 2D), and then using the Pythagorean theorem again with the base diagonal and the height to find the space diagonal.

Applying the Formula with Given Dimensions

The dimensions of the room are given as:

• Length ($l$) = 30 m
• Width ($w$) = 15 m
• Height ($h$) = 10 m

Now, we substitute these values into the space diagonal formula:

\( d = \sqrt{(30 \, \text{m})^2 + (15 \, \text{m})^2 + (10 \, \text{m})^2} \)

Calculating the Longest Pole Length

Let's perform the calculations step-by-step:

1. Square each dimension:

   \( (30 \, \text{m})^2 = 900 \, \text{m}^2 \)

   \( (15 \, \text{m})^2 = 225 \, \text{m}^2 \)

   \( (10 \, \text{m})^2 = 100 \, \text{m}^2 \)

2. Add the squared values:

   \( 900 \, \text{m}^2 + 225 \, \text{m}^2 + 100 \, \text{m}^2 = 1225 \, \text{m}^2 \)

3. Take the square root of the sum:

   \( d = \sqrt{1225 \, \text{m}^2} \)

   To find the square root of 1225, we can recognize that numbers ending in 25 often have square roots ending in 5. Let's try 35:

   \( 35 \times 35 = 1225 \)

   So, \(\sqrt{1225} = 35\).

Therefore, the length of the longest pole that can be placed in the room is 35 meters.

Comparing with Options

The calculated length is 35 m. Let's look at the given options:

• 31 m
• 33 m
• 35 m
• 18 m

Our calculated value of 35 m matches one of the options provided.

Revision Table: Key Geometry Concepts

Additional Information: Extending Geometry Problems

Understanding how to calculate the space diagonal is useful for various geometry problems involving 3D shapes. Here are some related ideas:

• Cube: A special case of a cuboid where length, width, and height are all equal (say, 'a'). The space diagonal of a cube is \(d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}\).
• Real-World Applications: This concept is used in engineering, architecture, and construction to determine maximum object sizes that can fit in spaces or for structural calculations.
• Pythagorean Theorem: The formula for the space diagonal is a direct extension of the 2D Pythagorean theorem ($c = \sqrt{a^2 + b^2}$) into three dimensions.

Remember that the longest object that can fit inside any rectangular container will always align with its space diagonal.