Final answer:
The temperature at which the infrared heater for the sauna must run to achieve the required power output is approximately 57 Kelvin.
Step-by-step explanation:
To calculate the temperature at which an infrared heater for a sauna must run in order to achieve a required power output, we can use the Stefan-Boltzmann law of radiation. The law states that the power radiated by an object is proportional to its surface area, the Stefan-Boltzmann constant, the object's temperature raised to the fourth power, and the object's emissivity.
In this case, we are given the surface area of the heater (0.050 m²), the required power (360 W), and the emissivity (0.84). Using these values and rearranging the equation, we can solve for the temperature of the heater.
Plugging in the values:
360 W = (5.67 × 10^(-8) J/s · m² K^4) * (0.050 m²) * (T^4) * (0.84)
Simplifying the equation and solving for T:
T^4 = (360 W) / ((5.67 × 10^(-8) J/s · m² K^4) * (0.050 m²) * (0.84))
T^4 ≈ 143,489 K^4
T ≈ 57 K
Therefore, the infrared heater for the sauna must run at approximately 57 Kelvin in order to achieve the desired power output.