Q:  What is the peak wavelength for a cool star of T = 3000 K?
            Which optical filter might show such stars best?


                  lambda_peak  =  967 nm   =~  1 micron

               This is at the very red end of the optical spectrum,
               closest to the I-band.



    Q:  Dust clouds can be warmed by starlight to temperatures of
            a few hundreds of degrees; let's use T = 300 K as
            a representative value.  What is the peak wavelength
            for light emitted by such dust?  What portion of the
            spectrum is that?

           
                    lambda_peak  =~  10 microns

                This is in the mid-infrared part of the spectrum.
                The Spitzer Space Telescope might see light of
                similar wavelengths in

                     IRAC 4   ( 8 microns)
                     MIPS 25  (25 microns)