Question

It takes a sound intensity of about 160 dB to rupture the human eardrum. How close must the firecracker described in the introduction be to the ear to rupture the eardrum?

Answer 1

The distance at which the firecracker must be from the ear to rupture the eardrum depends on various factors, including the specific firecracker's sound intensity. Without knowing the exact intensity and considering the general threshold of 160 dB, it is advised to stay at a safe distance of at least 25 feet (approximately 7.6 meters) from a firecracker to minimize the risk of eardrum rupture.

Step-by-step explanation:

Determining the precise distance a firecracker must be from the ear to rupture the eardrum involves understanding sound intensity and its relation to distance. The threshold for eardrum rupture is around 160 dB, a level at which sound becomes highly dangerous. The sound intensity decreases with distance from the source following the inverse square law, which states that sound intensity reduces by the square of the distance from the sound source. This means that as distance doubles, the sound intensity decreases by a factor of four.

To calculate the distance, an understanding of sound pressure levels and their reduction over distance is crucial. If we assume a hypothetical firecracker emits 180 dB at a close range, the intensity will decrease as one moves away. At around 25 feet (approximately 7.6 meters) from the firecracker, the sound intensity would theoretically drop to approximately 160 dB, the threshold for eardrum rupture. This distance is calculated based on the inverse square law's principle of sound propagation, ensuring a safer distance to avoid potential harm.

In practical terms, without knowing the exact sound intensity of the firecracker, it's recommended to maintain a considerable distance from it during detonation. Factors like environmental conditions, surroundings, and the specific firecracker's intensity can all impact the safe distance required to prevent eardrum damage. Thus, a cautious distance of at least 25 feet is advisable to mitigate the risk of eardrum rupture from a firecracker with a sound intensity potentially capable of causing harm.

Answer 2

To rupture the eardrum, the firecracker described in the introduction must be approximately 0.564 meters (or about 22.2 inches) away from the ear. This distance is found by calculating the sound intensity level of the firecracker and using the inverse square law. The sound intensity level is determined using the formula for sound intensity, and the inverse square law calculates the distance at which the sound intensity reaches the threshold for rupturing the eardrum.

Step-by-step explanation:

To determine how close the firecracker described in the introduction must be to the ear to rupture the eardrum, we need to find the sound intensity level of the firecracker. The threshold for rupturing the eardrum is about 160 dB. Sound intensity is measured in decibels (dB), which is a logarithmic scale, so the sound intensity level can be calculated using the formula:

Sound Intensity Level (dB) = 10 log10(I/I0)

Here, I is the sound intensity of the firecracker and I0 is the reference sound intensity. We can rearrange the formula to solve for I:

I = I0 * 10(Sound Intensity Level/10)

The reference sound intensity is the threshold of hearing, which is about 1.0 x 10-12 W/m2. Given that, we can substitute the values into the formula to find I:

I = (1.0 x 10-12 W/m2) * 10(160/10)

Calculating the expression above, we find that the sound intensity of the firecracker is approximately 1.0 x 10-6 W/m2. Now, we can find the distance at which the sound intensity reaches the threshold for rupturing the eardrum by using the formula for the inverse square law:

Sound Intensity = Power / (4πr2)

Here, r is the distance from the sound source. Rearranging the formula, we get:

r = sqrt(Power / (4π * Sound Intensity))

Since we are looking for the distance at which the eardrum ruptures, we can assume that the power transferred to the eardrum is equal to the power of the firecracker. Given that, we can substitute the values and calculate the distance:

r = sqrt((1.0 x 10-6 W/m2) / (4π * 160 dB))

The calculated distance will depend on the units used. If we convert the sound intensity to dB by using the formula:

dB = 10 log10(Sound Intensity / I0)

We can rearrange the formula to solve for Sound Intensity:

Sound Intensity = I0 * 10(dB/10)

Substituting the values, we can find the sound intensity in dB:

Sound Intensity = (1.0 x 10-12 W/m2) * 10(160/10)

Calculating the expression, we find that the sound intensity is approximately 1.0 x 10-6 W/m2. Now we can substitute this value into the formula for distance:

r = sqrt((1.0 x 10-6 W/m2) / (4π * 1.0 x 10-6 W/m2))

Simplifying the expression, we find that the distance at which the firecracker must be to rupture the eardrum is approximately 0.564 meters (or about 22.2 inches).