An antenna's input resistance splits into two parts, and efficiency is simply the useful share.

\(R_{in}=R_{r}+R_{L}\)

Here \(R_{r}\) is the radiation resistance — a fictitious resistance that accounts for the power actually launched into space — and \(R_{L}\) is the ohmic loss resistance of the conductors and nearby dielectric, which merely heats them. Since both carry the same current,

\(\eta=\dfrac{P_{rad}}{P_{in}}=\dfrac{I^{2}R_{r}}{I^{2}\left(R_{r}+R_{L}\right)}=\dfrac{R_{r}}{R_{r}+R_{L}}\)

Substituting the numbers :

\(\eta=\dfrac{67}{67+5}=\dfrac{67}{72}=0.9306=93.06\%\)

which is option 1.

Sanity-check the distractors. Option 2, 7.4 %, is the loss fraction 5/67 — the complement, and the classic slip of inverting the ratio. Options 3 and 4 correspond to no natural combination of the two numbers at all. A quick estimate settles it anyway: the loss is small compared with the radiation resistance, so efficiency must be close to 100 %, and 93 % is the only candidate.

Why 67 Ω is a realistic figure. A half-wave dipole in free space has a theoretical radiation resistance of about 73 Ω, which is why 75 Ω coaxial cable is the natural feeder for one; a slightly shortened dipole comes out a little lower, as here.

Where efficiency really bites is with electrically small antennas. Radiation resistance falls as the square of the length,

\(R_{r}=80\pi^{2}\left(\dfrac{L}{\lambda}\right)^{2}\ \Omega\)

so a short whip may have \(R_{r}\) below 1 Ω while its loss resistance stays at several ohms — efficiencies of a few per cent are common in AM broadcast and HF mobile work. The related figure of merit is gain, which is efficiency times directivity: \(G=\eta D\). Directivity is set by the radiation pattern alone, so it is efficiency that separates the two.

Hence, the efficiency is 93 %.