αmax, αmin definitions and output amplitude numerical

Definitions

Attenuation (loss) is $\alpha(\omega) = -20\log_{10}|T(j\omega)|$ dB, so a large $\alpha$ means a small output.

 alpha(dB)
     |                     ______________
amin |- - - - - - - - - - |
     |                   /
     |                  /  transition
amax |~~~~~~~~~~~~~~~~ /
   0 +---------------+---+---------------> w
       passband      wp  ws   stopband

• $\alpha_{max}$ ($\alpha_p$): the maximum attenuation allowed anywhere in the passband (e.g. 0.5 dB). It sets the allowed ripple or droop.
• $\alpha_{min}$ ($\alpha_s$): the minimum attenuation required everywhere in the stopband (e.g. 40 dB).
• Half-power frequency $\omega_{3dB}$: the frequency at which output power falls to half of its maximum, i.e. $|T(j\omega)| = |T|_{max}/\sqrt{2}$, an attenuation of 3.01 dB.
• Bandwidth: the width of the passband. For a low-pass filter it is $0$ to $\omega_p$ (or to $\omega_{3dB}$ for the half-power bandwidth); for a band-pass filter $BW = \omega_2 - \omega_1$, where $\omega_1, \omega_2$ are the lower and upper half-power frequencies.

Half-power frequency and bandwidth of a low-pass filter:

 |T|
 1.0    |------.
        |       `.
 0.707  |- - - - - -o   <- half-power point (3 dB down)
        |            `.
        |              `-.___
      0 +----------+-----------> w
        |<--- BW --->
                   w3dB

Numerical

Attenuation in dB is defined as

$$\alpha = 20\log_{10}\frac{V_{in}}{V_{out}}\ \text{dB} \;\Rightarrow\; V_{out} = V_{in}\times10^{-\alpha/20}$$

At $f$ = 30 kHz, $\alpha$ = 90 dB and $V_{in}$ = 1 V:

$$\begin{aligned} V_{out} &= 1 \times 10^{-90/20} = 10^{-4.5} \\ &= 3.162\times10^{-5}\ \text{V} \end{aligned}$$

Answer: output amplitude at 30 kHz ≈ 31.6 μV.

(The 78 dB at 20 kHz applies only to a 20 kHz input; a 1 V signal at 20 kHz would come out as $10^{-78/20} = 126\ \mu$V.)