Rectangular Waveguide
A rectangular waveguide is a type of transmission line used for guiding electromagnetic waves. It consists of a rectangular metal tube with a dielectric material filling the inside. The dimensions of the waveguide determine its cutoff frequency, which is the maximum frequency at which the waveguide can effectively transmit signals.

Calculating the Cutoff Frequency
The cutoff frequency of a rectangular waveguide can be calculated using the formula:

fc = (c / 2) * sqrt((m/a)^2 + (n/b)^2)

Where:
- fc is the cutoff frequency
- c is the speed of light in a vacuum (3 x 10^8 m/s)
- m and n are the mode numbers (integers representing the mode of propagation)
- a and b are the dimensions of the waveguide

In this case, the dimensions of the rectangular waveguide are given as 3 cm x 1.5 cm. To calculate the cutoff frequency, we need to determine the mode numbers.

Determining the Mode Numbers
The mode numbers (m and n) depend on the type of propagation mode in the waveguide. The dominant mode for a rectangular waveguide is the TE10 mode, which has a single wave propagation direction along the longer dimension of the waveguide.

For the TE10 mode, the mode numbers are:
- m = 1
- n = 0

Substituting these values into the formula, we can calculate the cutoff frequency:

fc = (3 x 10^8 / 2) * sqrt((1/0.03)^2 + (0/0.015)^2)
= (1.5 x 10^8) * sqrt(33.33 + 0)
= (1.5 x 10^8) * sqrt(33.33)
≈ 1.5 x 10^8 * 5.77
≈ 8.65 x 10^8 Hz
≈ 865 MHz

Converting the cutoff frequency to GHz, we get approximately 0.865 GHz. None of the given options match this value, so there may be an error in the question or the options provided.