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Is 155-dB ultrasound in the range of intensities used for deep heating? Calculate the intensity of this ultrasound and compare this intensity with values quoted in the text.
Question by OpenStax is licensed under CC BY 4.0.
Final Answer

$3.16 \times 10^3 \textrm{ W/m}^2$.

Yes, this intensity is between the range used for deep heating of $10^3 \textrm{ W/m}^2$ and $10^4 \textrm{ W/m}^2$

Solution Video

OpenStax College Physics Solution, Chapter 17, Problem 73 (Problems & Exercises) (1:41)

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Video Transcript

This is College Physics Answers with Shaun Dychko. In the textbook, we're told that intensities of ten to the three watts per square meter to ten to the four watts per square meter are used for deep heating with ultrasound. And we're told that this particular sound in this question has a sound level of 155 decibels. And so, we're going to have to figure out what is the intensity that corresponds to the sound level, and then see if that intensity is within this range, ten to the three to ten to the four. So, the sound level is ten multiplied by the logarithm of <i>I</i> over <i>I naught</i>, and we'll solve this for <i>I</i>, and then substitute in <i>beta</i> and figure out what <i>I</i> is. So, we'll divide both sides by ten and move this logarithm to the left side in order for the unknown to be on the left. And, we have logarithm <i>I</i> over <i>I naught</i> equals <i>beta</i> over ten. And then, we'll make both sides exponents of ten, and on the left side, we're left with the argument, which is <i>I</i> over <i>I naught</i>. A number raised to the power of a logarithm with the same base just gives us the argument of the logarithm. And on the right side, we have ten to the power of <i>beta</i> over ten. We'll then multiply both sides by <i>I naught</i> to solve for <i>I</i>, and we have a formula now for intensity in terms of intensity level. So, the reference intensity is ten to the minus twelve watts per square meter, and then we multiply that by ten to the power of 155 decibels over ten, and this is 3.16 times ten to the three watts per square meter. And yes, this is between ten to the three and ten to the four, and so it can be used for deep heating.