Answer
Although the temperature is doubled on the Celsius scale, the absolute temperature on the kelvin scale is not doubled. Therefore, the radiation rate would not increase by a factor of 16.
Work Step by Step
The radiation rate is proportional to $T^4$, where $T$ is the absolute temperature measured in kelvins.
Although the temperature is doubled on the Celsius scale, for example from $20^{\circ}C$ to $40^{\circ}C$, the absolute temperature on the kelvin scale is not doubled. In this case, the absolute temperature increases from $293~K$ to $313~K$. Therefore, the radiation rate would not increase by a factor of 16.
