Answer
$\lambda_{max} = 976~nm$
Work Step by Step
We can use Wien's law to find the wavelength at which the background radiation was then most intense:
$\lambda_{max} = \frac{2898~\mu m\cdot K}{T}$
$\lambda_{max} = \frac{2898~\mu m\cdot K}{2970~K}$
$\lambda_{max} = 0.976~\mu m$
$\lambda_{max} = 976~nm$
