Effective aperture and directivity are two ways of stating the same thing, linked by the wavelength :

\(A_{em}=\dfrac{D\lambda^{2}}{4\pi}\)

Step 1 — find the wavelength.

\(\lambda=\dfrac{c}{f}=\dfrac{3\times10^{8}}{100\times10^{6}}=3\ \text{m}\)

Step 2 — substitute.

\(A_{em}=\dfrac{30\times3^{2}}{4\pi}=\dfrac{270}{12.566}=21.48\ \text{m}^{2}\)

which is option 2. Note that the other three options are the same digits with the decimal point moved — the arithmetic must be carried through carefully rather than estimated.

What effective aperture means. It is the area from which a receiving antenna appears to collect power:

\(P_{received}=A_{em}\times\text{power density}\)

and it need not resemble the antenna's physical size at all. A half-wave dipole is a thin wire of negligible area, yet with \(D=1.64\) its effective aperture is \(0.13\lambda^{2}\) — more than a square metre at this frequency. For a dish, by contrast, the effective aperture is roughly 55 to 70 % of the physical area, the shortfall being the aperture efficiency.

The \(\lambda^{2}\) dependence is the important physical content. At a fixed directivity, a lower frequency means a larger wavelength and hence a far larger capture area — which is why long-wave antennas are physically huge and why microwave links can use small dishes. Rearranged the other way, \(D=\dfrac{4\pi A_{em}}{\lambda^{2}}\) shows that a fixed aperture gives more gain as frequency rises, the principle behind every satellite dish.

The relation is a reciprocity result, valid for any antenna in transmission or reception, and it is what makes the Friis transmission equation possible.

Hence, the maximum effective aperture is 21.48 m².