Question

The intensity, I, of a sound wave is defined as its power output per unit area and is measured in watts per square meter. A sound wave's intensity can be calculated using I = 870π²a²f², where a stands for the wave's amplitude in meters, and f stands for its frequency of vibration in hertz. If a particular sound wave has an intensity of 0.05 watts per square meter and an amplitude of 3 x 10⁻⁵m, which of the following is most nearly its frequency?

a) 8.77 hertz
b) 60 hertz
c) 6,500 hertz
d) 4 x 10⁷ hertz

Answer 1

The frequency of the sound wave, with an intensity of 0.05 W/m² and an amplitude of 3 x 10⁻⁵m, is most nearly 8.77 kHz, so the correct answer is (a) 8.77 hertz.

Step-by-step explanation:

The intensity, I, of a sound wave is the power output per unit area, measured in watts per square meter (W/m²), and can be expressed by the formula I = 870π²a²f², where a is the amplitude in meters, and f is the frequency in hertz. Given the intensity of 0.05 W/m² and an amplitude of 3 x 10⁻⁵m, we can rearrange the formula to solve for the frequency f:

f = √(I / (870π²a²))

Plugging in the values, we get:

f = √(0.05 / (870π²(3 x 10⁻⁵)²))

After calculating, the frequency f is most nearly 8770 hertz or 8.77 kHz, so option (a) is closest to the correct answer.