Expensive amplifier A is rated at 280 W, while the more modest amplifier B is rated at 45 W. What output wattage amp would sound about twice as loud as the 45-W amp?

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Understand the relationship between perceived loudness and power: The perceived loudness of a sound is proportional to the logarithm of its power. Specifically, the human ear perceives a doubling of loudness when the power increases by a factor of approximately 10^(1/10) ≈ 1.26.

Set up the equation for the power of the amplifier that would sound twice as loud as the 45-W amplifier. Let the new power be P_new. The relationship is given by: \( P_{\text{new}} = P_{\text{old}} \times 10^{1/10} \), where \( P_{\text{old}} \) is the power of the original amplifier (45 W).

Substitute \( P_{\text{old}} = 45 \ \text{W} \) into the equation: \( P_{\text{new}} = 45 \times 10^{1/10} \).

Simplify the expression: Calculate \( 10^{1/10} \), which is approximately 1.26, and multiply it by 45 to find the new power.

The result of this calculation will give the power output of the amplifier that sounds about twice as loud as the 45-W amplifier.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Decibel Scale

The decibel (dB) scale is a logarithmic way of measuring sound intensity. An increase of about 10 dB typically corresponds to a perceived doubling of loudness. This means that to sound twice as loud, the power output must increase significantly, not linearly, due to the logarithmic nature of the scale.

Sound Power and Loudness Relationship

Sound power is the total energy emitted by a sound source per unit time, measured in watts. However, human perception of loudness does not increase linearly with power; instead, it follows a logarithmic relationship. Therefore, to achieve a perceived doubling of loudness, the power must increase by a factor of about 10 times the original power.

Amplifier Ratings

Amplifier ratings indicate the maximum power output an amplifier can deliver to speakers. In this context, amplifier A at 280 W and amplifier B at 45 W suggest that A can produce significantly more sound power. Understanding these ratings helps in determining how much louder one amplifier can be compared to another, especially when considering perceived loudness.