Solve the variation problems in Exercises 77–82. The pitch of a m...

Question

Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?

Accepted Answer

Hello today we're going to be solving a word problem by using a proportionality constant. So this is what's given to us. The time it takes for an auditorium to be cleaned varies inversely as the number of people doing the cleaning. So that means that the time it takes for somebody to clean an auditorium is inversely proportional to one over end. And what this is saying is that for one task it takes this amount of people to do that task. Alright. So how can we use this? Well let's go ahead and keep reading. We're told that if it takes 75 minutes to clean it by two people, how long will it take for the auditory to be cleaned by five people? So we're gonna let t equal to the amount of time it takes to complete that task. We're gonna let n equal to the number of people. And we're looking for some type of proportion between two different scenarios. We are told that the first one is going to be two people and the second one is gonna be five. So we need to introduce some type of proportionality constant K. And doing so is going to give me the equation. T. is equal to K. Times one over N. So we're just going ahead and adding this K to this inverse proportionality. Alright, so how do we use this Exactly? Well we're tasked to find the amount of time it will take for five people to clean the auditorium. So we know that in that scenario and it's going to be five because that's the amount of people that we're going to need. Well if we plug that into the equation right now we get T. Is equal to K times 1/5. But we can't solve for T. Does just yet because we need to find a value for K. So how do we do that? Well in order to do that, we can go ahead and use the information given to us in the first scenario, the first scenario again restating is that it takes 75 minutes for two people to clean the auditorium. So in that scenario we're letting t the amount of time equal to 75 minutes and we're letting end the number of people equal to two. We can use these two values to solve for proportionality constant. So plugging this in directly to our equation gives us 75 is equal to K times 1/2. And in order to solve for K, I'm gonna multiply both sides of this equation by two. Doing so on the right hand side is going to cancel out this fraction and we are left with K is equal to 75 times, two 75 times two gives us the value of 150. So now we can use this proportionality constant and plug it into the scenario where we have five people and now we have a value for K. We have a value for N. We can use these two values to figure out the time. So T. Is going to equal to 100 and 50 times 1/5. That also means that when we multiply this together we're gonna get T. Is equal to 1 50/5 which is equal to 30. So that means that it's going to take five people 30 minutes to clean this auditorium. And that's gonna be our solution. Which also means that the solution to this problem is going to be C. So I hope this video helps you in understanding what the proportionality constant does. And I'll go ahead and see you all in the next video.

Watch next