Solve the variation problems in Exercises 77–82. The distance tha...

Question

Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?

Accepted Answer

Hello today we're going to go ahead and solve a word problem using proportionality ease. So the word problem tells us that it is known in physics that an object in free fall drops at a distance that is directly proportional to the square of the time it falls. If a vase drops from a tall building at a distance of 50 ft in 2.1 seconds, at what distance will it drop? In 5.6 seconds. So let's just go ahead and point out one thing that is given to us in the beginning, distance is directly proportional to the square of time. So just to write that down, we can go ahead and let a variable d represent distance and we know that distance is directly proportional to time squared. So we're gonna let a variable t equal time. Now we want to go ahead and find a proportion of distance between 2.1 seconds and 5.6 seconds. So in order to make an equation for distance, we're gonna need to introduce some type of proportionality constant. So we can let distance equal to k, which is going to be our proportionality constant times times squared. So how does this equation help us? Well, now that we have a proportionality constant, we can go ahead and get a value for this when distance is and time is 2.1 and use it to figure out what the distance will be in 5.6 seconds. So let's go ahead and just solve for our proportionality constant In this portion of the work problem, we are told that our distance is 50 ft and our time is 2. seconds. So again distance is equal to 50. Time is equal to 2.1. So let's go ahead and plug in these values to our equation so that we may solve for a proportionality constant. D is going to be 50 which is equal to K times 2.1 squared. Now, in order to Sulfur K, I'm gonna go ahead and divide both sides of this equation by 2. squared. Doing so is going to leave me with K equal to 50 divided by 2.1 squared. And this is going to be approximately equal to 11.34 with respect to two decimal places. Okay, so now we have a value for a proportionality constant. K is approximately equal to 11.34. So let's go ahead and use this to figure out what the distance will be when time is equal to 5.6 seconds. So in this case here, time is equal to 5.6 and we want to solve for distance d. So we're just gonna take these values and plug it directly into our equation from earlier doing so, distance is going to equal to 11. times 5.6 squared and again 5.6 is squared because time is squared, the approximate value of time Is going to be 31.36. So d is equal to 11. times 31.36. And multiplying both of these values together is going to give me an approximate value of 355. ft. Now keep in mind all of our solutions are with respect to one decimal place. So if we go ahead and round this to a single decimal place, D is going to be equal to 355.6 ft. And that's gonna be the solution for this problem. That also means that the solution to this word problem is going to be be 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.

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