Is there a theoretical limit to the order of the spectrum one would be able to observe and calculate? Justify your answer mathematically.

Yes, there's a limit

Explanation:

Yes, there's a limit which is set by nλ = 90°, because a transmission grating cannot transmit backwards!

In diffraction gratings, the formula is;

nλ = d sinθ

For the maximum number of orders produced,θ = 90°

Thus;

nλ = d sin(90°)

nλ = d

Hence, nλ < d, or n < d/λ

This will represent the highest order of transmission spectra that is observable.

However, a transmission grating will also be able to produce a reflection spectrum which is generally very dim, and not used much. In this, the orders are numbered from the reflection normal, and also limited to nλ < 90°

none, its a machine

there is no correct list among the choices you've provided.

the list might look something like this:

-- baby human

-- piccolo

-- flute

-- soprano singer

-- trumpet

-- tenor singer

-- trombone

-- baritone singer

-- baritone horn

-- bass singer

-- tuba

since i had to make up the list myself, i made it easy on myself, and i didn't try to shuffle the string instruments and woodwinds into it too.