Final answer:
To produce the same sound intensity, a 12" speaker has to move less than an 8" speaker due to its larger area. The correct answer, calculated based on the area ratio of the two speakers, is A) 0.67".
Step-by-step explanation:
The question at hand involves understanding the relationship between a speaker's size and its movement to produce sound of a particular intensity. According to the principles of physics, specifically acoustics, the power of a sound wave is proportional to the square of its amplitude. This means that if a smaller speaker must move a greater distance to produce the same energy as a larger speaker, then for a 12" speaker to produce an equally powerful sound as the 8" speaker moving 1" in amplitude, it can move less because of its larger surface area.
To calculate this, you can consider the area of the speakers. The area of a circle (which speakers typically are) is calculated by A = πr². An 8" woofer (which is a diameter of 8 inches) has a radius of 4 inches, and a 12" speaker has a radius of 6 inches. The area of the 8" speaker is thus π(4)² and the area of the 12" speaker is π(6)². This means the 12" speaker has 2.25 times the area of the 8" speaker. Since power is proportional to amplitude squared, and to produce the same power, the 12" speaker needs to move an amplitude which is the square root of the inverse of the area ratio. Hence, the 12" speaker should move by √(1/2.25) times the amplitude of the 8" woofer.
Doing the math:
√(1/2.25) = √(1/2.25) * 1" = 0.67"
Therefore, the correct answer is A) 0.67".