Other option: take the derivative of the area function (-4x+4000) and set to zero. The slope is zero at the vertex and it still yields X=1000!

@1234larry1

13 күн бұрын

Yes, to me its easier than finding the vertex and substituting it in.

@yodaami

12 күн бұрын

Yes I did that. But It’s effectively what he did when he used the formula. This is for pre calculus maths. The kids must be being taught the formula before learning differentiation.

@Patrik6920

12 күн бұрын

@@yodaami its the solution of x = - b/2a ± Δx, wher Δx=(±√(b²-4ac))/2a more generally known as the quadratic formula (-b±√(b²-4ac))/2a .. without the Δx term

The biggest area, if any shape was allowed, would be a semicircular fence.

@ocayaro

13 күн бұрын

😊 but not for the 99%

@tedschaft2785

13 күн бұрын

Can you prove that? Seems like it might be part of an ellipse.

12 күн бұрын

Would have been my answer too. Nowhere is stated that it has to be rectangular.

@yodaami

12 күн бұрын

Erm it is stated very clearly that it has to be a rectangle. Watch again.

@stevekru6518

11 күн бұрын

My screen says “largest rectangular area possible”

I'm completely distracted by this guy's ability to write backhanded!

Love these videos. Excellently explained!

That's exactly the real life example my Algebra teacher used except it was a barn instead of a river. 😄

@garys5175

14 күн бұрын

that is a BIG barn

The most efficient use of fencing to capture maximum area, in a four-sided enclosure, is always a square. When one side of the enclosure is provided, and you have to construct only three sides, the most efficient way is to create part of a square. The answer to this problem is given as half of a square. To answer these problems, the total length of the short sides always equals the length of the long side, or y=X1+X2, and total fencing = 2(X1+X2). It's also true that you can rotate the square that your half square is taken from. The long side stays the same length, anything reduced from one of the short sides is added to the other. The area remains the same. At the extreme, one side goes to length zero and the other increase to length Y. Now the river and the fencing form an isosceles triangle with fenced sides Y. The area is 1/2 (2000)(2000) is 2,000,000 m^2 and the length of fencing is 4000m. If you do this, you elimanate the need for one of the fence posts. That's even more efficient.

@carultch

15 күн бұрын

Fence posts for chain link fencing are generally needed every 10 ft, so you'll still need approximately the same number of fence posts no matter what shape you build. There is a little more expense for corner and gate posts, due to a need for a concrete footing and a larger post to withstand the extra load, but you'll still have a number of posts as a function of the length, whether it's a corner or a line post.

@mikefochtman7164

13 күн бұрын

If the corners require extra components to brace the corner post against being pulled over (often a diagonal brace from the top of the corner post downward along the fence to the base of the next post), this saves the 'extra equipment' of a corner. Another method if you don't really care about the shape so much, is fence in a semicircle with the river along the diameter. Or a 'semi-polygon' with each vertex a fence post. If there are enough, we might not need the extra bracing at each vertex and thus maximize the enclosed area.

@deakzoltan2714

13 күн бұрын

I like this idea of rotating the square. :) My first thought was a half circle (before I saw it was supposed to be a rectangle), and so the half square actually makes perfect sense to me. However, I do not really see, how you knew it must be a half square in the first place?

@Rohan4711

12 күн бұрын

​@@deakzoltan2714Just like you instantly knew the biggest area with a certain circumference is a circle the biggest area of a rectangle is a square. And you instinctively wanted to place the river as the diagonal of the circle it's exactly the same with the square.

Take the 4000m of fencing, and roll it into a ball. Then use the Banach-Tarski paradox to double the amount of fencing; repeat endlessly until you have infinite fencing. Then take a 1L Gabriel's horn filled with paint, and since the 1L of paint can clearly fill the 1L Gabriel's horn that has infinite surface area, use that 1L of paint to paint the infinite amount of fencing. I hope that this helps. ;-) !!

eGreat problem, but I remember this problem from calculus where you take the first derivative of the area function. Thanks for sharing!!!

Is a semicircle more optimal for area quantity? It sure would be for utility.

No river has straight banks so Step one - build it on inside of a meander of the river ... if it is a full ox bow and comes back on itself the enclosed area could be huge. Lateral thinking before math ...😊🇦🇺

I was always bad at this 😂 if a math problem deals with shapes in any way then it’s definitely harder for me to solve.

If Tom wasn't so fussy, he could fence in a semicircle with the river along the diameter, for an area of 2,546,479 (more than 25% more area). But that's outside the problem's restriction.

@MrBerryK

13 күн бұрын

That was exactly my first idea, but then I saw “rectangular area.” Oh well.

@Rohan4711

12 күн бұрын

Happy that Tom realized he wants a rectangular fence. Look at fences in the countryside and notice they are rectangular in shape. Why is that? Do build one rectangular fence and a circular one with similar rigidity and you will understand why you don't want a circular one. Then add the wasted land as you can never buy a circular plot of land.

How long would the sections have to be if you left a two meter space in front for a wooden gate?

Another way to think about it is to consider the case where he does need to fence all sides, which would be a square of side length 1000m. If we start with that square and remove the river side, we now have 1000m of fence to add to other sides to make the area bigger. To increase X by 1m takes 2m of fence. To increase Y by 1m takes 1m of fence. These values are fixed, considering the rectangular constraint.* Therefore, we increase our area the most by only increasing Y, leading to the same 1000 x 2000 area as the algebraic solution.

Geerkens' Law: For maximum area, total vertical fencing = total horizontal fencing = one half total fencing. Thus maximum area is 2000m * 1000m = 2,000,000 m^2.

You can fence a lot more area if you set up the fence in a semicircle.

There is a simpler way to analyse this problem that touches the heart of algebra: the observation that problems that can be described by the same equations have mathematically identical solutions irrespective of contexts, and therefore, it makes sense to analyse equations separated from contexts. It is well known that among all rectangles with a fixed perimeter, the largest area has a square. It can be proven using the multiplication formula (x-a)(x+a) = x^2 - a^2. It follows that the maximum of z*(b-z) is at z = b/2. The problem above can be formulated by a similar set of equations, y=4000 -z, 2*Area = z*(4000 -z) by substitution z=2x. Hence, z=2000, and x=1000, without knowing anything about parabolas.

@garys5175

14 күн бұрын

But the answer isn't a square

@KhanAcademyPoPolsku

14 күн бұрын

@@garys5175 It is not because the context is different. Nevertheless, a maximal value of the expression z*(b-z) is for z=b/2, so if you manage to transform your equation into this, you immediately get the answer, which you then have to transform into your variables, z=2x.

Mr H, could you please teach us extraneous roots?

Sir can you tell where did this formula come from? (Formula to calculate the largest area possible).

@carultch

15 күн бұрын

It's a formula he developed himself, based on the situation given. The area of the rectangle is x*y. The perimeter of the rectangle is 2*x + 2*y. Since one of the y's is the river, and we don't need a fence along it, we're only interested in the fence length of 2*x + y, which equals 4000 ft. So this generates the constraint equation: 2*x + y = 4000 And the equation we're trying to maximize, A = x*(4000 - 2*x) Since this is a parabola, maximizing it is as simple as finding the vertex, which is as simple as averaging the two x-intercepts. In a general sense, this would be an application of calculus, to solve for where the slope (dA/dx) equals zero.

@abhirajanita9640

15 күн бұрын

@@carultch thanks for the explanation

Sir we may use AM-GM inequality (2x + y)/2 >= √(2x * y) So after silving the inequality 2000000 >= xy Hence largest area is 2000000m²

Simple. Make the fenced area an half circle of radius 4000/π = 1273.2m, for an area of 2 546 479.1m. The problem shows a river, and don't forbid non-straight fences. There is nowhere that state that the fences MUST be straight.

@Rohan4711

12 күн бұрын

The task specifically stated the largest rectangular area. If that was not stated the largest area would be a half circle. To bring the problem slightly closer to reality I do like that they specified it should be rectangular. If you started to sell circular plots of land you would end up with a mess as there would be lots of land in between the circles. Also the practicality of putting in fence posts is important. You can set up quite a few posts in a straight line without the need of support. You often only need to add support for the corner posts, and with close to 90 degree angle the supports are easy to set towards the closest posts. With a circular setup you are in for a major problem unless you can have a super stiff fence pushing between posts. A normal fence is done with fence wire. A wire or rope is great for pulling force, but a disaster for pushing force.

When doing calculations it is easy to make misstakes, so in the real world you need some kinf of extra checking and convince others that you are correct before starting the huge task of setting up 4000 m of fence. Explaining that the solution presented in the video is correct to people that normally sets up fence posts could be a challenge. So I am suprised that noone suggested the iterative stupid-simple approach. Set a value for X, calculate Y and X*Y. Set a new value for X....se what gave the biggest area and repeat. Everyone will understand the math, and can see what X value resulted in the largest area.

So much easier with a bit of calculus.

4 kilometers of straight rivet? Oh, it's a hypothetical river.

@wbrehaut

13 күн бұрын

Make that 2 km?

@johnmckown1267

12 күн бұрын

@@wbrehaut yeah. my brain isn't as good as it used to be. I'm aging like fine milk.

Well if you only want to be a foot away from the bank😂

Dislike the use of a formula. Surely it is better to use the symmetry and largest rectangle will be the x value mid way between the roots. Roots are 0 and 2000 so x =1000. No remembering formula necessary.

Maybe I'm stupid but I can't see how he could enclose an area that large with 4000 metres of fencing. The largest possible rectangle will be a square, surely? Each side of that square will have a length of 4000m/3, so 1333.3'm, giving an overall area of 1,777,777.7'm. Please explain how he could enclose 2,000,000m using the avaiable fencing while sticking to the rectangular rule. I don't get it.

@mrhtutoring

10 күн бұрын

It's rectangular shape but the problem states that there's no fence on the river side.

@willemslie

10 күн бұрын

@@mrhtutoring Which is why I have divided the length of the fence into three sides, not four. You have not answered my question.

I liked this video before I saw it

And in the real world, you can't put a side fence in where the maths says it should go, because it happens to be where a river bed is, which is best avoided. But, it is OK another 20m away. Welcome to the world of constraints.

ME: Oooohhh, yeah!!! Totally get it, thank you Me after walking away: 2 million?? Did anyone else get that??

It's a square. 1000 metre x 1000 metre

@wernergamper6200

14 күн бұрын

No it’s not

1000x1000m

@wbrehaut

13 күн бұрын

Did you miss that the river must be one side of the rectangle?

@leondupreez1123

12 күн бұрын

Aaaa!!

The largest area that is possible, is when x=y and we can certify that x•y=4000 m^2 4000^1/2=63,245 x=y=63,245

@user-hq4gu1jb3n

15 күн бұрын

???? First, you're forgetting a few zeros, which you do redeem, but your max as x=y only applies when there are four sides - i.e. two "x"s and two "y"s. We don't have that here, so the x=y doesn't apply. Note that the actual answer is equivalent to two squares adjacent to each other of 1000 metres a side.

@davidnewell3232

15 күн бұрын

X and Y are sides of a square. The total length is 2x+2y or 4x or 4y. Therefore the maximum total area possible, when building all sides of an enclosure with 4000m of fencing is when each sides is 1000m. This means the maximum area is 1,000,000 m^2. Using a natural boundary for one side, doubles potential area. To build all four sides of a 4000m^2 pen with fencing, in the most efficient way, would mean that each side should have length = (4000^1/2) m in length. That means the sides would be 63.246 m long and the total length of fencing 252.98 m.

@ernesthakey3396

8 күн бұрын

​@@davidnewell3232 your math is extremely wonky and wrong. If you have 4000 m of fencing, and you use that to make 3 sides of a square, then each side will use exactly 1/3 of those 4000 m, ie 4000/3 m. So in that arrangement, X=Y=4000/3. Therefore the area of the square is (4000/3)*(4000/3)=1,777,777.777..... We can round that up to 1,777,778 square meters. However, that is not the maximum area we can get. If we consider instead that the edge of the river bisects a larger square that is 2000 m on a side for what would be a full square using 8000 m of fencing, knowing that we have only 4000 m of fencing, what we're really solving is what is the largest half of a square we can get. As the video shows, if the long side is 2000 m and the two short sides are 1000 m, then the area we enclose as half a square is 2,000,000 square meters. 2,000,000 sq m > 1,777,778 sq m.

I don't get it. The 4000 is irrelevent. You are showing that 2*x = y is the answer for all such problems.

@wbrehaut

13 күн бұрын

No. The question was, "What's the largest rectangular area possible?" The answer must be an area in square meters, which "2x=y" is not. And your question seems to be implying that the area wouldn't depend upon how much fencing your have available.

An easier way is to create a square with 1000 meter sides. Take the 1000m length from the riverside / 2 = 500 and add 500m to each end of the square bottom, slide the square verticals to each end of the new line making it a shape 2000m long x 1000m wide. 2000 x 1000= 2,000,000 sq m

@garys5175

14 күн бұрын

that's crazy

@BangkokBubonaglia

13 күн бұрын

That may be easier, but you haven't explained why it works. Maybe you could have gotten a bigger area dividing that riverside fence by 3 and adding to each of the 3 sides. You don't actually know until you've done the analysis above, showing that this is the correct solution. Then it's easy to see alternate ways to visualize it.

"Using algebra in real life" means on the SAT/ACT. 😃

I would do a circle and fence of 2.546.479 square meters. That's 25% more!

Um,,, a circle…

What about half-circle ?

@wbrehaut

13 күн бұрын

"What's the largest rectangular area possible?" A circle?

@Ivan-fc9tp4fh4d

13 күн бұрын

@@wbrehaut I know. But it should be a "real example" and the goal was "max area". :) The best would be a bend of the river with radius 2000...

@Ivan-fc9tp4fh4d

13 күн бұрын

@@wbrehaut I know. But it should be a "real example" and the goal was "max area". :) The best would be a bend of the river with radius 2000...

It annoys me that people use "meters square" and "square meters" interchangeably.

@jamesharmon4994

14 күн бұрын

Here is the difference: A square plank with sides of 2 meters is a plank that's 2 meters square, but the area is 4 square meters.

I can stand this sort of phrasing. "What's the largest rectangular area possible?" You mean, in the universe or what? If you mean "What's the largest rectangular area possible that Tom can enclose with his fence?" then write that! You want me to do the calculation properly, have at least the curtesy to phrase the problem properly. It's not only pettiness but it will also avoid confusion. Mind you, this is not addressed at Mr. H but at whoever phrased this. (Unless it was Mr. H, then it will be addressed to him! 😅)

@user-hq4gu1jb3n

15 күн бұрын

Well, the question, along with the diagram is perfectly plain. I don't see how it could be any plainer without being pedantic.

@alphalunamare

15 күн бұрын

@@user-hq4gu1jb3n It works on The Rio Grande so no worries :-)