Solution to this University Admission rectangle problem (TMUA 2023)

[00:00:02]Because the perimeter is 40 and that's a square, so each side is just equal to 10. And we're given that this right here is equal to X. And the goal is to make sure that this rectangle here has area centim- 20 cm squared.

[00:00:20]So, how do we do it? Well, first thing first, this right here, the shape has two lines of reflection, no symmetry.

[00:00:29]If you look at this, well, it has to be this symmetric line and this symmetric line, right? So, if you know this right here is X, this right here will also be X.

[00:00:44]Okay.

[00:00:45]So, how do we do this though? Well, you have X and X, so that means this is a 45-45-90 special right triangle. So, that means this side here has to be square root of 2 * X.

[00:01:00]Good.

[00:01:02]Now, if you look at this right here, because the whole thing is 10, from here to here has to be just 10 - X.

[00:01:14]So is this.

[00:01:19]So, that means again, we have a 45-45-90 special right triangle. So, this right here has to be square root of 2 * 10 - X for this side.

[00:01:32]And now we have a rectangle with Oops, that's that's a that's not the right angle notation, anyways.

[00:01:43]This is the side and that's the side we know. So, we know the area of the rectangle is equal to square root of 2 * 10 - X * square root of 2 * x.

[00:01:59]And we are going to set this equal to 20.

[00:02:07]And then we will just work this out. So, this times this is 2 and then we have x * 10 - x. That's equal to 20.

[00:02:17]Divide both sides by 2 first, then we will also distribute this.

[00:02:21]I divide by 2 like this.

[00:02:24]So, we get 10x - x² = 10.

[00:02:30]And if you bring this to the other side, it's not factorable. And because we know the coefficient of x is even, so we'll complete the square.

[00:02:38]However, this right here is -x², right?

[00:02:42]So, I'm going to divide -1.

[00:02:46]And we get x² 10x. That's equal to -10.

[00:02:54]Now, to complete the square, we have to take half of the coefficient of x, which is -10.

[00:03:02]And then square that.

[00:03:03]That's -5², which is 25.

[00:03:08]So, we add 25 here and 25 here.

[00:03:12]This is factorable and we get x - 5 squared.

[00:03:16]And that's 15.

[00:03:18]And then do the usual business.

[00:03:22]And then add 5 to both sides. So, x is equal to 5 plus or minus square root of 15.

[00:03:29]And now notice that we have both plus or minus.

[00:03:33]In fact, both of them give you positive x.

[00:03:36]So, we cannot rule out any of that.

[00:03:37]However, the question is asking for the largest x value that will give us this right here, 20. So, we are going to pick the larger value. So, the largest x is that.

[00:03:52]5 + square root of 15.

[00:03:55]Just because the question is asking for the largest X.