Question

In order to make flat boards from a log, a miller first trims off the four sides to make a square beam. Then the beam is cut into flat boards. If the diameter of the original log was 15 inches, find the maximum width of the boards. Round your answer to the nearest tenth.

Short answer

The maximum width of the boards is 10.6 inches.

Step-by-step solution

Understand the Problem

We're given a circular log with a diameter of 15 inches. The miller trims the log into a square beam. We need to find the maximum width of the boards cut from this square beam.

Visualize the Log and Square

The diameter of the log indicates that it's circular with a diameter of 15 inches. When the miller trims the log to make a square, this square's diagonal equals the diameter of 15 inches.

Use the Diagonal of the Square Formula

For a square, the relationship between the side length (let's call it \( s \)) and the diagonal is given by the formula: \( d = s\sqrt{2} \). Here, \( d = 15 \), the diagonal of the square.

Solve for the Side Length of the Square

Using the formula \( d = s\sqrt{2} \), we substitute \( d = 15 \) to find \( s \):\[15 = s\sqrt{2}\]Solve for \( s \):\[s = \frac{15}{\sqrt{2}}\]\[s = \frac{15 \times \sqrt{2}}{2}\]

Calculate the Maximum Width

Compute the value of \( s \):\[s = \frac{15 \times 1.414}{2} \approx 10.61\]Round \( 10.61 \) to the nearest tenth: \( 10.6 \). Therefore, the maximum width of the boards is 10.6 inches.

Key concepts

Diagonal of a Square

The diagonal of a square is a crucial concept in geometry, especially when dealing with shapes inside circles or creating squares from circular objects. When you have a square, the diagonal is the line segment that connects two opposite corners of the square. It holds a special relationship with the sides of the square. You can find the length of a diagonal using the formula:

• \( d = s\sqrt{2} \)

where \( s \) is the side length of the square and \( d \) is the diagonal.
This formula derives from the Pythagorean theorem, which applies to right-angled triangles. In the context of the problem, applying this formula helps in determining the square cut out from a circular log.
If the diagonal (which equals the diameter of the circle) is known, in this case, 15 inches, you can rearrange the formula to solve for the side length \( s \), enabling you to convert the circular shape into a manageable square.

Trigonometry

Trigonometry, though not directly mentioned, plays a vital role in understanding the principles behind calculating the diagonal of a square. Trigonometry explores relationships in triangles, particularly right-angled ones. When dealing with diagonals and squares, you inherently deal with a right triangle formed by two sides of the square and the diagonal.
The trigonometric relationship primarily provides the foundation for the diagonal formula. It's based on the Pythagorean theorem, which suggests that in a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (sides of the square).

• \( (s^2 + s^2 = d^2) \) simplifies to \( (2s^2 = d^2) \)

This insight into triangles and trigonometry underpins our ability to calculate lengths and angles in such geometric problems.

Mathematical Applications

Mathematical applications in real-world scenarios often involve using geometry and algebra together, as in this beam-cutting problem. Understanding how to convert from one shape to another, like a circle to a square, is a common task.
When you're trying to maximize certain aspects, such as the width of boards from a log, you apply these mathematical concepts practically. By knowing how the diameter of a circle relates to the diagonal of a square, you can precisely determine the largest possible square—or in this case, the maximum width of boards obtainable.
This exercise demonstrates the real-world applications of math in fields like construction or forestry, where optimal material usage is crucial. These applications showcase the importance of mastering geometry and algebra, as they help solve practical problems efficiently and help students recognize math's value beyond the classroom. These principles also guide decisions on resource utilization, ultimately affecting outcomes like cost-efficiency and material waste.