Ed has 120 ft of fencing to fence in a rectangular plot. What is the maximum area he can enclose?
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- เผยแพร่เมื่อ 31 ม.ค. 2024
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Once again a confusing and poorly stated problem - the problem states "around a rectangular plot" which means ALL 4 SIDES of the rectangle. The fact it is next to a river means nothing UNLESS you mean no fence on the river side in which case you are talking about 3 sides of the plot will be fenced, but not the side along the river. So which is it - fencing in all 4 sides of the plot or just 3 sides.
Your observation raises the essential element of engineering: clarifying the REQUIREMENTS of a problem or project.
I thought it was perfectly clear. The plot is rectangular shaped (the plot does, indeed, have 4 sides) with one side being the river. Thus you only need fencing for 3 sides.
@@fleetadmiralj Definition of 'around' - "located or situated on every side". The question states "he wants to put a fence around a rectangular plot" which means the fencing goes on all four sides.
The problem is not ambiguous and the river is irrelevant. Fencing around means all the way around.
@@ellentronicmistress4969 alright. come up with a way to word it so that it doesn't use "around" or some other term that wouldn't imply to someone being pedantic that the fence goes on all four sides. My initial thought was "put a fence enclosing a rectangular plot" but I can immediately see people having the same problems with that wording as well, because the point here is to insist there is ambiguity when it is perfectly clear what the question is asking.
River doesnt count as enclosed. If he puts animals in there they can either drown or swim across and escape. Or an unwanted neighbor can swim across. Answer is 900.
Agree. If one can assume the river counts as a fence, you can also assume the plot will be used to keep animals.
In the owner's criteria, the river suffices.
The problem should state that the riverbank is one of the sides of the rectangle, and that the shoreline is perfectly straight. Still 👍
When I read the problem I thought the statement about the river was extra information
I think there are two possible solutions to this depending on whether you are using the river as footage to enclose the property. So you can get two answers
In case of a test I would opt for two answers with explanation.
Ok, first I didn't understand that the river is one of the 4 sides of the rectangle.
Given that, I'd say 1800ft², length of the lot is 60 ft, width is 30 ft to get the maximum area possible.
Wrong or misleading…..the key word being “around” your fence doesn’t go “around” the rectangle.
Exactly.
Imagine my amazement (when I was in my 20's) when I discovered by accident that the formula for the area of a sphere is the derivative of the formula for the volume of a sphere!
Problem offers a good review in using differential calculus.
"Rectangular plot" and "enclose" are key words, a rectangle has four sides not three, enclose means all the way around, not open on one side, the correct answer is 30x30= 900 square feet
If you assume that a square also is a rectangle.
OK. Does this guy ever read the comments so he can learn from us especially when the problem is not worded well?
No.
got it by brute force - 1800 good one. knowing that formula would make it easier.
If 1800sq' is the largest area possible, then what are the length and width of that rectangle? I cannot find it. 30'x30'=900sq' - (this rectangle happens to be a square and seems to yield the greatest area). Let's progressively increase the length and decrease the width: 35'x25' = 875; 40x20 = 800; 50x10=500; 55x5=275; 59x1=59. It appears that a square (shaped rectangle) is the max size area at 900sq'. Am I losing my mind?
At 7:14 he says 'X+X+length = 120'. It should be X+X+2(length)=120.
At 7:55 he says 'length = 120-2x' . It should be length= (120 - 2X)/2.
I got lost as the explanation expanded but it seems that he confused total distance of both "length" sides added together, with the (single) length side used in formula A=LxW.
Yes. The river is one side of the rectangle, so you don't need fencing for the 2nd "long" side. That's why it is 120 - 2x. x is the length of the shorter side, which we need two of, but we only need one "long" side because the other "long" side is the river.
@@fleetadmiralj I understand. However, the problem John posed would have been clearer if it said the rectangle only has 3 sides (the river being the 4th side). This is not intuitive. Most rectangles I have seen usually have 4 sides.
These are some of my favorite kinds of problems. Did it using calculus. Was wondering if you were going to explain. You did not disappoint. Actually wasn’t aware of the first equation you used from PC but was able to derive it using calc.
lol this was me exactly. Got it using the calculus solution. Was unaware of the vertex equation. (actually got it by trial and error first but the calculus solution confirmed it)
I need one more dimension...the length of a single property line. The problem is, if I make the fence line longer, with the depth shorter, that reduces effective area for the same perimeter. IF...I can use the river as a perimeter line, and, I'm allowed to make a square, the total area would be a maximum of 1,600 square feet. If I move to a rectangle, the area will be less, but, I do need one more value. I do understand, that if I change the aspect ratio by even a fraction of an inch, it technically becomes a rectangle. the final answer would be
Thanks again John you explained it visually as well,now I get it.
Assuming he wants a rectangular plot fenced in, the answer is 900 sf, unless the plot of land is either less than 30 ft long or less than 30 ft wide.
If he wanted to fence in the most land possible, and his plot was long enough and wide enough, he could put up a circular fence and get a fenced-in area of about 1147 sf.
Well it would be a circle of perimeter 120 foot. So about 1146 square feet if it was flat. The plot he wants to enclose and the river do not impinge on the actual question, which is clearly stated. (A river cannot enclose anything that is higher than it) Beyond that, with a bit of landscaping he can increase the surface area of the land he is enclosing, so he should be able to enclose maybe 1400 square feet.
As it happens this will also fulfil the farmer's wish to enclose a rectangular plot, since there are many rectangles of enclosed land within the fenced area.
Exactly, it doesn’t state the size of the plot. If it’s a small plot he won’t need the 120ft of fencing, but ok let’s find the maximum. It also says “to fence in” in the title and in the problem it says “to put a fence around”, both of which indicate the fence being unbroken and enclosing the entire maximum area. A river does not imply that one side of the plot is fenced in or that the plot has been fenced around. In my day that was considered extraneous information and would have been dismissed as not part of the problem. I’m actually stunned by his solution.
This is probably completely the wrong way to do it but ..
120 ft of fencing would give us a square 30 ft by 30 ft and a square would be the largest area because both the length and the width would both be the biggest number possible.
But .. we don't need one side of the square here because there is a river here .. and if we think of the answer as a rectangle, made up of two squares next to each other, we won't need the side between the two squares either, giving us enough fencing to do a 60 ft by 30 ft rectangle .. giving us an enclosed area of 1800 sq ft.
That is actually a pretty smart way of deriving the answer without using the fomulas he used.
What you started at 16:40 was unnecessary. Once you have x (the width) all you have to do is double that and subtract from 120 to get the length (60 ft.). The 30 x 60 = 1800.
But it was nice to see the use of the formula to its conclusion.
Calculating area is easy but the problem is confusing. I assumed "enclose" meant all the way around and the plot being next to a river is irrelevant. Hence, 900 sq. ft. One side open is not enclosed. If the plot were against a cliff, three sides would make it enclose.
Exactly, the wording is making it difficult to understand since you can get 2 different answers. I did the same thing as you originally before seeing others only use 3 sides to answer
I actually learned something here. Thx.
You said a rectangle “around” a plot of land. That says that the fence is continuous around totally enclosed on four sides. The river has no fence there fore the land is not enclosed by the fence, which you specified .
I found my error. I didn't account for the free extra perimeter gained from lengthening the corral, being allowed to use the river as a fence line while making it only slightly more narrow. I might have gotten that, IF I knew that the river could be used as a fence line. The square would be maximum area for a given perimeter length, if I was limited to the 120 linear feet of fencing. The river is 60 feet of free fencing, giving a total perimeter of 180 feet.
If instead of a river a cliff face was available your enclosure would be closed off. The action of leaving a side open to a water attack is a poorly designed fortress. With a cliff face uou only need to watch for an attack from above.
ok this comment made me laugh after all of the "but how could we know one side was the river!" comments
It would still be wrong though because the question states 'put a fence around'
"MANY MATH TEACHERS GOT IT WRONG!!" John doesn't get an A+++ or stars nor can he brag to friends and family. The question needs to be clearly stated in the title for those of us that attempt the problems first, then watch the video solution. "Ed has 120 ft of fencing to fence in a rectangular plot. What is the maximum area he can enclose?" The correct answer to the problem as written and read 3 times is 800 square feet.
How did you get 800?
I think perhaps you meant 900 sq ft?
@@ellentronicmistress4969 In that case the rectangle is a square. Okay, a square is also a rectangle, but I would go for a 'genuine' rectangle, so 899 square ft. (31 x 29 = 899). :)
@@Kleermaker1000 A square is a type of rectangle. End of.
@@ellentronicmistress4969 Formally you are right, as I already wrote, but it was obvious that John did not mean a square.
This assignment was mostly a semantic challenge. It was multibigous. Due to my limited knowledge of the English language, I did not know that a fenced, rectangular "plot" could have a fence only along three out of four sides of the rectangle. And by the way, the boundry between a river and land is never a straight line ;) The following answers should thus be considered correct: (120 feet/4)^2 (or 30 feet squared = 900 square feet) or "Ø". Now I have to watch the rest of the video to figure out the answer was 2 * 900 square feet ...
Well, I got the right answer, but more by trial and error (I figured 40 x 40 and 20 x 80 gave the same area so I figured the answer was in-between so went 30 x 60). Came up with 120x - 2x^2 as a formula but kind of was at a loss after that (hey, it's been 20 years since I've been in math class lol). I actually eventually got it using the calculus solution you used at the end. I was kind of surprised by the vertex formula because I don't actually remember that.
I'm sorry, but given the wording of the question, it's quite an assumption to make that the river is one boundary of the plot - you really need to state that fact. This is the kind of poorly worded question that can actually put young students off maths.
why is it quite an assumption? If a river doesn't form one side of the rectangle, there is no reason to even mention it.
@@fleetadmiralj In the title, 'to fence in' is defined as surrounding something completely with wood or wire. A river might be a boundary, but it certainly is not a 'fence' - at least not in my country (maybe the term has a looser meaning in the US). In the question on the board, it states 'put a fence around' which likewise suggests the fence goes completey around the plot: definition of 'around' - 'located or situated on every side'.
@@fleetadmiralj And exactly what shape is "a river?" Some have been known to have curves in them. The wording of this problem is terrible, that's the point here. It could have been stated as "For a given perimeter, what is the aspect ratio of the rectangle of maximum area?" The answer is "2 to 1."
@@noseyparker8130 I don't even know what to say. It's a math question about making a rectangle. If that isn't enough to make the assumption the river is straight I don't know what is. Otherwise you basically make any word math problem unsolvable because there is always going to be a "but what about"
Excellent
The river bank may be a curve, and we have no way of knowing that!
■ There is a playground next to a school, it is fenced around .. all 4 sides, so as to contain the soccer ball while a game is in progress.
■ My grandfather has a meadow NEXT TO A RIVER, where he rear horses. The meadow is fenced around, all 4 sides, so that the horses doesn't fall in the river.
●●● Sorry to say, but this Maths Teacher, must read his questions at least four (4) times to ensure that it is not AMBIGUOUS !!!
These ambiguities are becoming far to prevalent.
120 ft / 3 sides = 40 ft so the area is A = 40² = 1600 ft²
Ed has a rectangular plot and wishes to fence in the maximum area. He can enclose a larger area than the given solution by forming a semi-circle with a radius of 38.197 feet and an area of 2,291.83 sq.ft. This can be placed in a rectangle of 76.394x38.197. This is significantly larger - 27+% larger.
Once you get x=30, remember the x is the width, so length is 120-2(30)= 60 , and area is 60*30=1800
Once again an incorrectly stated problem. By definition a rectangle has 4 sides with the fence assumed to bound both sides. Stating the problem with the require that the fence go "around" requires that the fence complete a bounding circuit. The river does not count as a bounding side as your original premise states that the fence will "enclose" the area. If you wish to argue that the river indeed counts as a bounding side that I would propose putting up the fence at the mouth of a canyon with an unspecified length giving an answer than cannot be defined. The answer of a completed bounded fence will be a maximum of 900 sq ft.
y = -2x^2 + 120x is a quadratic function, not a quadratic equation.
If he wants to fence in only three sides (was not made clear in the wording of the problem), he could fence in a maximum of 1600 sf.
900 sq ft
2 sides at 40 ft each, 2 sides at 20 ft each
Oh, the federal government won't let that happen.
If he only fences three side, he did NOT enclose the rectangle, only three sides of it. The river side is NOT enclosed. So it is 900 Sq. ft. To QUOTE: "He wants to put a fence AROUND A RECTANGULAR PLOT...", EMPHASIS ADDED. NOT to put a fence around a "U" shaped area.
I se someone also saw this confusion
900 sq ft. I believe a square is a rectangle.
1 ,600sq ft.
I have the wrong answer
Poorly worded. Around a plot doesn’t mean three sides. My answer is 900 sf. ft, because a square is a rectangle by definition.
The river affects nothing.
We are tricked this time! :)
For a three-sided enclosure. Since the lengths 40 and 80 yield equivalent outcomes (40x40=80x20), the mean between the two represents the peak of a bell curve ...thus length 60 and sides 30 for 1800 sq ft.
Now I know why I said Screw math, it was from teachers like this, "UNSUBSCRIBED".
60'x30'x30'. 1800 sq ft
60x30x30=54000.
What he means is that the fence is 60 feet long and the two ends which are 30 feet each. The area of a rectangle is is length times width which is 1800 sq ft as MrWhipple1062 indicated
@@bulldog6925 Question just says "next to a river". (1) Why can't you have one side (of 4 sides) of the fence built along the river water line?, (2) What gomer is going to build a rectangular fence with 1 of 4 sides missing? Every farmer knows if you have one side of the fence missing, all the cows will run away. :))
@@terry_willis Nice, 3d-fencing !
@@bulldog6925
Mr whipple indicated incorrectly. 60’ x 30’ x 30’ = 54,000 cubic ft. That’s volume, not area. 60’ x 30’ = 1800 square ft. That’s the area.