Final answer:
After estimating the decimals for each plumber's fractional and irrational estimates, the lengths of the radii in order from the greatest to least are: Plumber Z (pi/24 inches), Plumber W (3/25 inches), Plumber X (3/11 inches), and Plumber Y (9/100 inches).
Step-by-step explanation:
To determine which plumber's estimate is the greatest and which is the least, we need to compare the fractional and the irrational values they provided for the radius of a cylindrical pipe. Since π (pi) is approximately 3.14159, we can use this approximation to help us rank the estimates.
Let's compare the estimates:
- Plumber W: ⅓ inches = 0.12 inches
- Plumber X: ⅓⁄₁₁ inches. Since 3 ≈ 0.27 for this purpose and 11 is larger than 25 in Plumber W's estimate, this radius is smaller than Plumber W's.
- Plumber Y: ⅓⁄ₙ₀ inches = 0.09 inches
- Plumber Z: π⁄₂₄ inches. Since π ≈ 3.14159, π⁄₂₄ ≈ 0.131 inches, which is larger than any of the other estimates.
Therefore, after converting each estimate to a decimal and comparing them, the lengths in order from greatest to least are:
- Plumber Z: π⁄₂₄ inches
- Plumber W: ⅓ inches
- Plumber X: ⅓⁄₁₁ inches
- Plumber Y: ⅓⁄ₙ₀ inches