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we're told in this question that on a string instrument the length of a string so let's call that L the length of a string L varies inversely as the frequency so varies inversely as the frequency so L is going to be equal to some constant times the inverse of the frequency times the inverse of the frequency and I'll use F for frequency the frequency of its vibrations and then they tell us the vibrations are what give the string instruments their sound that's nice and 11-inch and it's actually the vibe it's the vibrations of the string affecting the the air and then the air compressions eventually get to our eardrum and that's actually what gives us the the perception of the sound but we don't want to delve too much into the physics of it an 11-inch string has a frequency so 11-inch so this is its length so the 11 is string has a frequency of 400 cycles per second so this right here is the frequency a cycle per second is also called a Hertz find the constant of proportionality and then find the frequency of a 10-inch string so they say an 11-inch string so 11 so 11 inch string is equal to some constant of proportionality times 1 over 400 cycles per second so 1 over 400 cycles cycles per per second all right second is sec so to solve for the constant of proportionality we need to multiply both sides by 400 cycles per second 400 cycles per second multiply the left-hand side by 400 cycles cycles per second and the left-hand side becomes 400 times 11 well 4 times 11 is 44 so 400 times 11 is 4,400 and then we have in our units just for out of interest our units are cycles times inches per second so it's cycles cycles times inches in the later of our units divided by seconds and that is equal to our constant of proportionality so we can say we can say that the length the length is equal to forty four hundred forty four hundred times and or forty four hundred cycles cycles times inches per seconds I want to get the unit's right per second times one over the frequency times one over the frequency so we solved for a constant of proportionality and then we can use this to find the frequency of a 10-inch string so now we're talking about a situation where our length is 10 inches so 10 inches 10 inches so we get 10 inches 10 inches they are equal to 4400 as you can imagine these units are that a little cumbersome but 4,400 cycles times inches times inches per second times 1 over the frequency times 1 over the frequency and so we can do a couple of things we could just multiply both sides of this equation by the frequency so that it gets out of the denominator so let's do that let's multiply both sides by the frequency by the frequency and then we could also divide both sides by 10 inches to get rid of this and then we'll just have frequency on the left hand side so we divide both sides by 10 inches and then we divide it by 10 inches 10 inches the left hand side we're just left with the frequency we're just left with the frequency on the right hand side we have 40 400 divided by 10 4,400 divided by 10 is 4 440 and then you have in cycles inches over second divided by inches well the inches cancel out with inches and you are just left with cycles cycles per second and then obviously 1 over F times F cancels out to just being 1 so we get our frequency when our when our string is 11 inches long or when our string is 10 inches long our frequency has increased to 400 recycles per second from when it was 11 inches when was slightly longer our frequency is 400 cycles per second when our string got a little bit shorter one its shorter now it our frequency increased by 40 cycles per second