Step 1: Understanding the Question:
The question asks for a direct numerical calculation of the reverberation time of a specific hall given its volume and total absorption capacity.
This is a standard calculation in architectural acoustics to ensure clear speech and music quality.
Step 2: Key Formula or Approach:
We use Sabine's empirical formula for reverberation time in SI units:
\[ T = \frac{0.161 \cdot V}{A} \]
Where:
\(V = 5000 \text{ m}^3\)
\(A = 200 \text{ metric sabins}\)
Step 3: Detailed Explanation:
Step A: Identify the given physical parameters. The volume of the hall is quite large, at 5000 cubic meters. The total absorption, which accounts for the surface area and absorption coefficients of walls, ceiling, and floor, is 200 units.
Step B: Substitute the values into the formula.
\[ T = \frac{0.161 \times 5000}{200} \]
Step C: Simplify the arithmetic. First, divide 5000 by 200:
\[ 5000 / 200 = 50 / 2 = 25 \]
Step D: Perform the final multiplication:
\[ T = 0.161 \times 25 \]
Step E: Detailed multiplication:
\(0.161 \times 10 = 1.61\)
\(0.161 \times 20 = 3.22\)
\(0.161 \times 5 = 0.805\)
\(3.22 + 0.805 = 4.025 \text{ seconds}\).
Looking at the options, 4.025 s is most accurately rounded to 4 seconds.
A reverberation time of 4 seconds is generally considered very high for a lecture hall (where 0.8–1.2s is ideal) but might be acceptable for large cathedrals or specific musical venues.
Step 4: Final Answer:
The calculated reverberation time for the given hall is 4 seconds.