Sure, let’s solve the problem of finding the diameter of the circular window step-by-step:
1. Given Information:
- The horizontal shelf length is \(8 \text{ ft}\).
- The vertical brace length is \(2 \text{ ft}\).
2. Understanding the Geometry:
- The diameter of the circle coincides with the length of the horizontal shelf.
- The vertical brace acts as a radius and forms a right-angle triangle with half of the horizontal shelf.
3. Forming the Right Triangle:
- Half the shelf length is \(\frac{8}{2} = 4 \text{ ft}\).
- The brace length is \(2 \text{ ft}\).
4. Applying the Pythagorean Theorem:
- In the right triangle, the total radius of the circle formed is the hypotenuse.
- Let’s denote the radius by \(r\).
- The Pythagorean theorem states \(r^2 = (\text{half shelf length})^2 + (\text{brace length})^2\).
5. Substitute the Known Values:
- \(r^2 = 4^2 + 2^2\).
- \(r^2 = 16 + 4\).
- \(r^2 = 20\).
6. Solving for the Radius \(r\):
- \(r = \sqrt{20}\).
- \(r \approx 4.472 \text{ ft}\).
7. Calculating the Diameter:
- The diameter is twice the radius.
- Diameter \( = 2 \times r \).
- Diameter \( = 2 \times 4.472 \).
- Diameter \( \approx 8.944 \text{ ft}\).
Thus, the diameter of the window is approximately [tex]\(8.944 \text{ feet}\)[/tex].
