Final answer:
To find the power input needed for a speaker to produce a 90.0-dB sound intensity level, we convert the dB level to sound intensity, calculate the speaker's area, and use the speaker's efficiency to calculate power input using the given formulas and efficiency percentage.
Step-by-step explanation:
To calculate the power input needed to produce a 90.0-dB sound intensity level for a speaker with a given efficiency, we need to use the relationship between decibels (dB), power, and efficiency. The sound level in decibels can be converted to sound intensity (in watts per meter squared) using the formula:
I = I0 × 10^(level in dB / 10)
where I is the sound intensity, I0 is the reference intensity (10-12 W/m2), and level in dB is the sound intensity level in decibels. Using this formula, we find the sound intensity for a 90 dB sound.
Next, we calculate the area of the speaker's surface using the diameter provided. The area A of a circle is πr2, where r is the radius. The radius is half of the diameter; thus, for a 12.0 cm diameter speaker, the radius is 6.0 cm or 0.06 m.
Once we have the sound intensity and the area, we can find the power output (Pout) from the speaker by multiplying the sound intensity by the area (Pout = I × A). The efficiency of the speaker allows us to determine the required power input (Pin) using the efficiency formula:
Pin = Pout / (Efficiency / 100)
Finally, we input the efficiency value of 1.00% to find the power input needed to produce the specified sound intensity level right at the speaker.