Final answer:
The first-order maximum for the longest wavelength will occur at a larger angle than 30.0°, but the exact angle cannot be determined without additional information. It will be greater than 30.0° and less than 90°.
Step-by-step explanation:
The diffraction grating equation, given by d sin(\theta) = m\lambda, where d is the grating spacing, \theta is the diffraction angle, m is the order of maximum, and \lambda is the wavelength, can be used to determine the angle at which different wavelengths will produce maxima. Since the question gives us that the first-order maximum for the shortest wavelength occurs at 30.0°, we can infer that for longer wavelengths, the angle should be greater because the wavelength \lambda is directly proportional to sin(\theta). The longest wavelength of visible light is approximately 760 nm (red light), and since red light has a longer wavelength than the shortest wavelength of visible light, the first-order maximum will occur at a larger angle than 30.0°. Without knowing the exact grating spacing, we cannot calculate the exact angle, but logically, the longest wavelength should correspond to an angle larger than 30.0° but less than 90°, as 90° would imply a backlight situation which is not possible in a diffraction pattern. Options b) 45.0° and c) 60.0° are both possibilities, but without additional information, we can't determine between them. The most we can confirm is that the angle will be greater than 30.0° and less than 90°.