One cue your hearing system uses to localize a sound (i.e., to tell where a sound is coming from) is the slight difference in the arrival times of the sound at your ears. Your ears are spaced approximately 20 cm apart. Consider a sound source 5.0 m from the center of your head along a line 45° to your right. What is the difference in arrival times? Give your answer in microseconds. Hint: You are looking for the difference between two numbers that are nearly the same. What does this near equality imply about the necessary precision during intermediate stages of the calculation?

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Hey, everyone in this problem, we're told that cats employ sound localization techniques to catch mice. OK? So they can determine the direction of the mouse. Thanks to variations in the sounds that reach their auditory system. We're told to consider a mouse that's gnawing three m away from the center of the cat's head along a line 30 degrees to the left. And which we can see in the diagram here, we're asked to find the difference in the arrival times of the sound of night. We're told that the difference between the cat's left and right ears is seven centimeters. We're asked to assume a room temperature of 20 degrees Celsius. We're given four answer choices all in microseconds, option A 35 option B 62 option C 102 and option D 248. So what we're looking for is the difference in arrival time. OK. So we're gonna call that delta T that's what we're looking for and that's just gonna be equal to the time it takes to arrive at the right ear minus the time it takes to arrive at the left ear. OK? And in reality, we want the absolute value of this. OK. That difference, we're not sure which one is gonna be bigger. But from looking at the diagram, we expect that the right year will be a longer time because it's further away. OK. So we do expect that one to be bigger. So we shouldn't have to take the absolute value. Um writing it the way we've written it. Now, we aren't given any information about time. OK? We have some information about distances here. We know how far the mouse is away from the center of the ears. Now, let's recall that we can write the speed V is the distance D divided by the time T which tells us that the time T can be written as a distance D divided by the speed V. So in order to find these times T R and T L, the time to reach the right ear, the time to reach the left ear respectively, we need to find D and V for both ears. Nice, so fine today and beef for both years. And let's start with the distance. OK. So let's start with D R and D L. Now D R, we are gonna draw on this diagram and we are gonna do it in blue, the distance from the mouse to the right ear. OK. It's gonna be just this direct distance D R. And we can imagine if we fill in this triangle. Mhm What do we have? Well, let's first start with the 30 degree angle with the mouse to the center. OK. If we call the X distance, just X, OK. And we call this why distance, why we have this angle of 30 degrees? We know the hypotenuse is three m. So we can figure out what X and Y are. All right. So we're gonna do that in a minute, but I'm gonna show you why we need to do that. Let's look back at our blue dotted lines, right? If we find X, well, that's gonna be the exact same side length for the distance to the right ear. OK. So we have this one side length, the other side length is gonna be Y plus, this is a little bit extra and that little bit extra is half of the distance between the two ears. Maybe we know the distance between the two ears is seven centimeters. So this extra little distance is gonna be 3.5 centimeters. All right. So let's start on that year and then we'll move to the left ear when we're done. So starting with the calculation for X and Y, why is gonna be related to our angle 30 degrees through sight? OK. We know that the hypotenuse is three m and so Y it is gonna be equal to three m multiplied by sine of 30 degrees. OK? We're using sine because we're talking about the opposite side. Similarly, X is gonna be related through cosine because it's the adjacent side. Again, the hypotenuse is three m. So we get X is equals to three m multiplied by cosine of 30 degrees, right. We have our X values, we have our Y values. What that means is we know two sides of our blue triangle for this right ear. OK. We wanna find the hypotenuse, let's use Pha Green theorem. OK. That tells us that the distance D R squared is going to be equal to the first sideline squared, which is X squared plus the second side length squared, which is gonna be Y plus 3.5 centimeters. We wanna write this in meters because all of the other distances are in meters. So to convert from centimeters to meters, we divide by 100. So we have Y plus 0.35 m all squared. Now we know X and we know y let's substitute those in. We get that D R squared is equal to three m multiplied by cosine of degrees all squared plus three m multiplied by sin of 30 degrees plus 0.35 m all squared. And if we work this out on our calculator and we take the square root, we get that this distance to the right ear from the mouse, D R is gonna be 3.1765 m approximately. All right. So we found the distance. Er now we're gonna move to the left ear and then we're gonna do it in a very similar way. So we're gonna draw in green if we draw from the mouse directly to that left ear. Let me do that again. That's not so straight. OK. So we're gonna draw a straight line this time to the left ear. Yeah, we can imagine making this triangle and this triangle is going to be very simple. The X distance is still just X and the distance from the mouse to the left and right ears horizontally is the same vertically. OK. We had why? And now we're a little bit short of why. How short of why? Well, 3.5 centimeters right half of that distance between the two ears. And so our side length is going to be Y minus 3.5 centimeters. OK? For the right ear, it was plus 3.5 centimeters and we went to the center of the ears and then went a little bit further. The left side is the opposite came, we're a little bit short of that center. So when we write out our equation for D L using the Pythagorean theorem, again, you get that the distance to the left ear, D L squared is equal to X squared plus Y minus 0.35 m squared. Substituting in our values D squared is going to be three m multiplied by cosine of 30 degrees plus three m multiplied by stein of 30 degrees minus 0.35 m, all squid. And again, working this out on our calculator, taking the square root, we get that the distance to the left ear D L is gonna be 2.9826 by four m. So we have our two distances now, ok. Remember we're trying to calculate the difference in time to calculate time. We're doing the distance divided by the speed. We have the distances. Now, we need to find the speed and the speed is gonna be the speed of sound. And we're given the temperature in this room. OK. So we're called at the speed of sound. The sound is equal to 331 m per second, multiplied by the square root of the temperature in degrees Celsius T plus 273 divided by 273. And you might have seen this equation as just the square root of T divided by 73. OK? Instead of having this addition in the numerator, and that equation is exactly the same, but the temperature is given in Kelvin. OK. So because we have our temperature in degrees Celsius, we have to add 273 in order to convert it to Kelvin first. And so this form of the equation just takes into account that conversion. So in this problem, we have a room temperature of 20 degrees Celsius. So the speed of sound is gonna be 331 m per second, multiplied by the square root of 20 plus 273 divided by 273. And if we work this out, we get the speed of sound. In this case is 342. m per se. All right. So now we can get back to what we really want to find. And that's those times. So let's start with the time it takes for the sound to reach the right ear, that's gonna be the distance to the right ear divided by the speed of sound. V substituting in the values we just found 3.1765 m divided by 0.91 m per second, which gives us a time of 0. seconds. And if we do the same for the left ear, we get that T L is equal to D L divided by V which is equal to 2.98, 2654 m divided by 342.91 m per second for a time of 0. seconds. And one final step. Remember this question was asking for the difference in the arrival times. And so again, delta T that difference in arrival times is gonna be T R minus T L and we're gonna subtract these two times. We found to get a difference of 0.1, 02 seconds. OK. Now our answer was in microseconds. The answer choices are in microseconds. So we wanna go ahead and convert this, OK? Microseconds. We want 10 to the exponent negative six. If we move our decimal decimal place six places 123456, it's gonna be after 102. So we can write this as 102 times 10 to the exponent negative six seconds. OK? Which is equivalent to 102 microseconds. And that's exactly what we were looking for. So we found that the difference in the arrival time up the sound for those two ears is 102 microseconds which corresponds with answer choice. C Thanks everyone for watching. I hope this video helped see you in the next one.

Verified video answer for a similar problem:

Key Concepts

Sound Localization

Interaural Time Difference (ITD)

Geometry of Sound Propagation