An FM modulator operates with output $x_c(t)=10sin(200t+(t)).$ The modulator operates with $f_d=2H\frac{z}{V}$ and the message signal $m(t)=5sin(20t).$ The modulator is followed by a bandpass filter with a center frequency of $100$ Hz and a bandwidth of $30$ Hz$.$

b$.$ Determine the modulation index.

c$.$ Determine the power at the filter input and the filter output.

d$.$ Draw the single$-$sided amplitude spectrum of the signal at the filter output.