What is meant by the bandwidth of a transmission medium? What is its importance?

In amplitude modulation, the message signal $V_{\mathrm{m}}\left(t\right)=10\sin \left(2\pi \times {10}^{5}t\right)$ volts and carrier signal $V_{\mathrm{C}}\left(t\right)=20\sin \left(2\pi \times {10}^{7}t\right)$ volts. The modulated signal now contains the message signal with lower side band and upper side band frequency, therefore the bandwidth of modulated signal is $\alpha \mathrm{kHz}$. The value of $\alpha$ is: