Final answer:
The number of nodes in a standing wave that is three wavelengths long is 7, including the endpoints. By counting the nodes at the ends of each wavelength and considering the fixed nodes at the boundaries of the wave, the total is derived.
Step-by-step explanation:
The number of nodes, including the endpoints, in a standing wave that is three wavelengths long is 7. Each full wavelength of a standing wave has 2 nodes at its ends. Since the wave is three wavelengths long, it would have nodes at the beginning and end of each wavelength. However, the nodes at the end of one wavelength and the beginning of the next are the same. Thus, for three wavelengths, you have 2 nodes for the first wavelength, plus 1 node each for the next two wavelengths (since the end node of one wavelength is the beginning node of the next), yielding a total of 4 nodes.
Now, remember we have nodes at both ends of the entire wave (since it is a standing wave and the boundaries are fixed). Since we've already counted these when considering the nodes for each wavelength, we do not need to add them again. Counting all distinct nodes we get 4 (for wavelength divisions) + 2 (one at each end) = 6, plus the one at the very start, so we end up with a total of 7 nodes. Therefore, the correct answer to this question is D. 7.