The perimeter of a rectangle is 26 meters and its area is 40 squa...

Question

The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.

Accepted Answer

Hello today we're gonna be fighting the dimensions of a rectangle. So the problem tells us to consider a given rectangle. It has an area of 221 square meters and a perimeter of 60 m. We want to find its length and width. So we're dealing with a rectangle with an unknown length and width. We can assign length as X. And width as Y. And we're dealing with the area and the perimeter of a rectangle. So the area of a rectangle as represented as length times width and the perimeter of a rectangle is represented as two length plus two width. Now we can go ahead and come up with two equations using the area and perimeter of a rectangle. We are told that the area is m2. So we can replace a with 221 and that's going to be equal to length, times width or in this case x times y. And let's go ahead and come up with an equation for perimeter for a perimeter, we are told that the perimeter is going to be 60 m. So we could replace P with 60 and that's going to be equal to two length plus two with or two X plus two. Y. And if you notice 62 and two are all factors of two. So we can go ahead and divide the entire equation for r perimeter by two. To simplify it, dividing everything by two is gonna give me an equation. 30 is equal to X plus Y. And these are gonna be the two equations. We're gonna use to find the dimensions of this rectangle. So how do we start this process? Well we can't use the system of equations because both of these equations have different operations, but that means that we can go ahead and use the substitution method. So if we come up with a substitution for X from our area equation we can plug that into our perimeter and find the dimensions of why? So let's go ahead and do that in order to sulfur X in our area. We just want to divide both sides of this equation by Y. Doing so is going to give me X. Is equal to 221 over Y. Now what I'm gonna do with this is plug this substitution in To our perimeter equation. Doing so is going to give me 30 is equal to X. Which is 221 over y plus y. Alright so now what I wanna do is go ahead and solve for Y. But if you notice I have a Y. In the denominator here. So we need to get rid of that denominator first. So let's go ahead and isolate this value 221 over Y. In order to do that. We're going to subtract Y from both sides of this equation and that's gonna give me 221 over Y. Equal to 30 minus Y. Now what I can do with this is go ahead and multiply both sides of this equation by why to get out of that denominator. Doing so is going to give me equal to why times 30 -Y. Now I'm gonna go ahead and distribute this y into this parentheses. Doing so is gonna give me 221 equal to y minus y squared. And since we want to solve for y let's go ahead and set this equation equal to zero. In order to do that, I can add y squared to both sides of this equation and I can subtract 30 Y from both sides of this equation. Doing so is going to give me why squared minus 30 Y plus 221 equal to zero. Now this is a fact arable, Trin Amiel what this factors to is why minus 17 times why minus equal to zero? And solving both of these for why is going to give me the values why is equal to 17 and Y is equal to 13. Okay, so we have two values for Y. But which one do we use to help us find our dimension for X. Well we can go ahead and plug in both both values for y into our area to solve for X. So if we plug in Y is equal to 13, What we get is 221 equal to x times y. Which is 13. Now, in order to solve for X, I'm gonna, I'm gonna divide both sides of this equation by 13, so X. Or this equation divided by 13 is gonna give me a value of X is equal to 17 and the dimensions for this is going to be 13 by 17 m. Now we want to go ahead and check the value for Y is equal to 17, ideally if we plug this in, this should give us the same dimensions. So when y is equal to 17 and we plug it into our area, We're gonna go ahead and get 221 equal to x times y, which is 17. Now, what I'm gonna do is go ahead and divide both sides of this equation by 17. Doing so is going to give me X is equal to and the dimensions are gonna be 13 x 17, so exactly the same as when Y is equal to 13. So the length and width is gonna be 13 x 17. And so the answer to this problem is going to be a so I hope this video helps you in understanding how to find the dimensions of a rectangle and I'll go ahead and see you all in the next video

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