B

Step 1: Use the Stefan-Boltzmann Law for radiation heat transfer, which is given by the equation:
\( Q = \sigma A (T^4 - T_s^4) \)
where:
- \( Q \) is the rate of heat loss (in Watts),
- \( \sigma \) is the Stefan-Boltzmann constant (approximately \( 5.67 \times 10^{-8} \, W/m^2K^4 \)),
- \( A \) is the area of the emitting surface (assumed constant),
- \( T \) is the absolute temperature of the metal piece,
- \( T_s \) is the absolute temperature of the surroundings.

Step 2: Given that at \( T = 1400 K \) the heat loss rate is 200 J/s (or 200 W), we can express this as:
\( 200 = \sigma A (1400^4 - 300^4) \)
Step 3: Now, we need to find the heat loss rate at \( T = 800 K \):
\( Q' = \sigma A (800^4 - 300^4) \)
Step 4: Calculate \( 1400^4 \), \( 300^4 \), \( 800^4 \):
- \( 1400^4 = 3841600000000 \),
- \( 300^4 = 810000000 \),
- \( 800^4 = 409600000000 \).
Step 5: Substitute values to find \( Q' \):
\( Q' = \sigma A (409600000000 - 810000000) \)
Step 6: This simplifies to:
\( Q' = \sigma A (408790000000) \)
Step 7: Now takes the ratio of heat losses:
\( \frac{Q'}{200} = \frac{(800^4 - 300^4)}{(1400^4 - 300^4)} \)
Step 8: Calculate the value:
First calculate \( 800^4 - 300^4 = 408790000000 \) and \( 1400^4 - 300^4 = 3841600000000 - 810000000 = 3840790000000 \).
Step 9: So the ratio is approximately \( \frac{408790000000}{3840790000000} \approx 0.1064 \)
Step 10: This indicates the new rate of heat loss:
\( Q' \approx 200 \times 0.1064 \approx 21.28 \, W \)
Round to significant figures gives approximately 21 W.
Therefore, the answer is 21 W.