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The owncr of # moicl has 30XH} m of fencing and wants t0 enclose rectangular plot Of land that borders straight highway: If she docs not fence tie side along the hi...

Question

The owncr of # moicl has 30XH} m of fencing and wants t0 enclose rectangular plot Of land that borders straight highway: If she docs not fence tie side along the highway, what is the largest area that can be enclosed?

The owncr of # moicl has 30XH} m of fencing and wants t0 enclose rectangular plot Of land that borders straight highway: If she docs not fence tie side along the highway, what is the largest area that can be enclosed?



Answers

Enclosing the Most Area with a Fence A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

Okay in this situation a farmer has 2000 m worth the fencing. Um and he's going to put a fence, rectangular fence nearer highway. Um but not using the fence along the highway. So here's my terrible drawing of a highway. The green parts, the fencing. If we think about this logically alright if you're making a rectangle and you don't know the lengths or the widths um the ones that are across from each other have to be the same. So I'm going to use X in that spot. I had 2000 m of fencing but when I subtract away the fence that I already used, Which is those exes um this remaining side it can be 2000 -2 XS that I had. All right. You know, once I kind of have a diagram for it, I can use what I know about rectangles in in their area. Well the area of a rectangle is length times with okay, so in our scenario our length would be X. In our with would be 2000 minus X. All right, so if I distribute that X. And I'm actually going to um right mind into standard form hopes I should say 2000 minus two X. Um I'm going to write mine in standard form right away X. Times negative two X. Um is negative two X squared X. Times 2000 would be plus 2000 x. All right. Now that I have my quadratic. Um For the scenario the questions asking me you know what's the largest area um that I can enclose with this fencing anytime they're asking you for maximum value. Um When it comes to quadratic, they're talking about the vertex and if you think about it logically the graph of this function would be a downwards facing parabolas. So if you want the maximum value of it, you're looking for the vertex that the highest point of it. All right. So we need to use our vertex formula which is the opposite Um that X is equal to the opposite of B over two times the a. That's my vertex formula. All right. So the opposite of B would be negative 2000 over two times the ace over two times negative two. All right. So this is negative 2000 divided by -4. Which is going to equal 500. All right. So now that we know that X is equal to 500. Even though I wrote 5000 I can use that to substitute back into the formula to find my maximum area. All right. Um so just to finish this out, I'm subbing in negative two and instead of XI'm plugging in my 500 squared plus 2000 Times 500. And you can put this into a calculator. Um you can do the math out in your head. That's fine. Um but this is going to come out to be um 500,000 meters squared because you're talking about area

So you have 2000 m of Fantine and you want to include rectangular plot that borders the highway. It's straight highway. You don't want to fence that along the highway. So what's the largest area that can be enclosed? Known this information then we would have a rectangle that's going to have side lengths of X. And then 2000 minus two X. To represent our sides. So that means our area is going to be equal to X times are 2000 minus two X. We then need to distribute the X. So it's going to be equal to 2000 X minus two X squared. We now have a quadratic and we know what R. And R. B. R C zero. So A. Is a negative two and B is going to be 2000. And we know that our maximum area is going to be at a maximum X value at the X value. Where is the maximum on a quadratic equation on the graph which the max son is going to be, X is equal to opposite of be all over to a. So plugging in our information, we know that our B is 2000. And since this opposite, it's going to be negative all over our two times or a which is negative two. We now have negative 2000, divided by a negative for a negative 2000, divided by negative four is 500. That's going to be our X value where our maximum area is. So that means area is going to be equal to 500 times 500 solar area is going to be 250,000 m squared.

We have the highway right over here And we have 3000 m of fencing. So let's say the fan uh fencing looks something like this. So this side is X. Which means that side is X. And this side is why? So since we have 3000 m of fencing in total, this tells us two X plus Y is equal 2 3000. Now the area of this rectangle is going to be X times what? Okay. And we're trying to maximize this area. Okay, So we're going to do is we're going to use this equation here to isolate why and then substituted into this equation here. Okay. So meaning if we isolate why we get Y is equal to 3000 -2 x. So now A is only in terms of X. So this is going to be X times 3000 -2 X which is 3000 X minus two X squared. So if we want to maximize the area, we know that has to happen either at the end points or at where a prime of X is equal to zero. So let's take the derivative And we get this is 3000 -4 x. So if we set a prime of X equal to zero and solve for X, we get X is equal to 1500 over to and the end points would be when x zero. Uh Well why is zero, but if one of them is zero, then we know the area is going to be zero. So those can't be um maximum areas. So the largest area has to happen When when X is equal to uh 1500 overture. So therefore max area is equal to and we can just use this equation here. So max area is equal to a of 1500 over to, Which is 3000 times 1500 over two two times 1500 over two square. Okay, so that ends up being, so let's put that into a calculator minus two times. Um that's just 7 20. I don't know why I wrote it as a few 100 over to, so that's minus two times 7 50 squared. So that is equal to, is that right? You just double check. That's a big number. Let's see, that's one 125,000. So 1.125 million. And our units here with the senses an area, this is going to be meters squared. Yeah.

Your four Children were professing is given to include a rectangle area where former didn't friends One side off this rectangle lady. So from the given in Fort Mason, you can write to lend off this rectangle Aerial support is equal Toe X meter and Vic Toews. This rectangle will be ableto 4000 minus licks NATO because one side is not finished so area will be equal to form Love area. We know that it is lend multiplied with made So we'll put you lengthen Then area will be ableto x multiplier to it for Teligent minus Alexis. So we will get 4000 times x minus. Who works is great. No, We have find maximum area enclosed by this fancy. So we will hear maximum this area To maximise this area, we will differentiate this function off this x rated for area. This area will be equal to know year by D x Quito or Teligent minus forex. Now, to maximize this area, we will put where you the year by the X Equal Pajero. We can write for Teligent minus for external big toe Jiro. Then will you actually be equitable? One told you. And meter No let me have 1000 meter so they can find that will be ableto or Teligent minus. Who will tell Jin that will be pulled to who? Teligent meet that? No, we will find maximum area that will be equal to one. Told you multiplier to it. Who told you that will be when the luck. It's glad make that this is maximum area enclosed by this fancy.


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