Question

A sphere having temperature 600 K is losing heat due to radiation. At this temperature its rate of cooling is \(R\). The rate of cooling of this sphere at 400 K is \(\frac{x}{243}\). The value of \(x\) is (Temperature of surrounding is 300 K)

• A. \(50R\)
• B. \(45R\)
• C. \(40R\)
• D. \(35R\)

Hint:

Rate of cooling is proportional to T to the fourth minus T0 to the fourth for radiation.

Answer 1

The Correct Option is D

Step 1: Approach
Scale all temperatures by 100 K to keep the arithmetic small.

Step 2: Units of 100 K
Surroundings 3, hot body 6 and 4. The cooling is proportional to $T^4-3^4$.

Step 3: Values
At 6: $1296-81=1215$. At 4: $256-81=175$.

Step 4: Reduce
$\dfrac{175}{1215}=\dfrac{35}{243}$, so the new rate is $\dfrac{35}{243}R$ and $x=35R$. Option (D).

Final Answer:
The ratio of the radiation terms is 175 to 1215, which is 35/243, so x = 35R, option (D). \[ \boxed{35R} \]