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A farmer wants to make three identical rectangular enclosuresalong a straight river, as in the diagram shown below. If hehas 1,200 yards of fence, and if the sides ...

Question

A farmer wants to make three identical rectangular enclosuresalong a straight river, as in the diagram shown below. If hehas 1,200 yards of fence, and if the sides along theriver need no fence, what should be the dimensions of eachenclosure (in ft) if the total area is to be maximized?side perpendicular to the river= ftside parralell to the river= ft

A farmer wants to make three identical rectangular enclosures along a straight river, as in the diagram shown below. If he has 1,200 yards of fence, and if the sides along the river need no fence, what should be the dimensions of each enclosure (in ft) if the total area is to be maximized? side perpendicular to the river= ft side parralell to the river= ft



Answers

A farmer wants to make three identical rectangular enclosures along a straight river, as in the diagram shown below. If he has 1,200 yards of fence, and if the sides along the river need no fence, what should be the dimensions of each enclosure (in ft) if the total area is to be maximized? side perpendicular to the river=



ft side parralell to the river=



ft

So we know that the we have 1000 ft of fence and we're gonna have a rectangle and then we're going to divide it up into three fields like so and the fence is going to go all the way around and we're going to have these extra two down the middle so that we divide this up into three equal pieces. So if we let this whole dimension from here to here, B. X. In this dimension from here to here be wide, then we can see that our perimeter will end up being we'll have X plus acts, so we'll have to acts +1234 wise. And they're going to have to add up to 1000 ft a fencing that will put fencing through this whole all this diagram. So let's solve for y. So we have four. Y. Is equal to 1000 minus two acts and then dividing everything by four. So we know that that dimension why in terms of X is 250 minus half a backs. So now we want to deal with the area of this plot and the area. I'll call that A. Of X. Because I want the function written in terms of X is the product of X times Y. But X times and this is what Y is equivalent to. So that will be the area. Now this grabs a parabola that if we distribute that X through is going to have a negative X squared term. And so this problem is going to open downward and there's a zero at zero and there's a zero. If we let 2 50 minus one half X equals zero. That's 12 50 is equal to one half X. And multiplying both sides by two. That's when X is 2 50. So this area function has a zero at zero and another zero at 2 50. And so we know that that problem is going to look like. So I'm sorry, this is 500 minus take multiplying both sides by two. This is 500. Sorry about that, guys. And now we know that the middle that vertex of that parabola is going to be at 2 50. So that is where you will maximize the area with that 1000 ft. So we want the Parabola or the fencing to be 250 ft. Bye. And let's see what the other dimension, the other dimension is. Why. And we know that why is 250 minus half of X, which is half of 2 50. This is 1 25 to 50 minus 1 25 is 125 ft. So that we'll take on that dimension to 50 ft by 125 ft.

Okay. We have probably number 13, in which there is a farmer who wants to friends. His farm, which is in rectangular in shape, and one side of the farmer is along the river. So there is no need to fence Columbia River if this is the river. So he had to just He has to just France, Three sides, three sides of this, uh, from okay, if this is oh, why this is X. So this will be X and we have fencing 1800 ft, which means the perimeter of this side parameter of this rectangle field X plus y plus X equal to 1800. So why will be equal to 1008 ended? Minus tricks. Now we have to maximize the area, maximize the area so area will be equal to accent away because this is a rectangular field. So let's plug in the value of X value of y from here xn to 1800 minus two weeks. Okay, so it will be the function of X, so x will be equal to 1800 X minus two x esquire. Okay, so we have to just maximize and minimize. Uh, here we are to maximize. So we need to get critical points To get the critical points, it s X should be equal to zero, So 1800 minus four X should be equal to zero. So for it should be equal to 1800. Okay, so X will be equal to 1800 by four. That is 450 feet. So this there is only one critical point, which is X equal to 450 ft. We have to check if this critical poor and corresponds to minimum or maximum. So a double dash X, that is differentiate this. It will be minus fel minus four, which is negative. So we have maximum here. Okay, so we are getting the maximum area at X equal to 450 ft, which was the work. And so why will be equal to 1800 minus two into 4. 50. That is 900 ft. So this is for under 50 ft. This is 900 ft. So these are the dimensions to get the maximum media

In this question, we have to find what dimension will heal and closer off the largest possible area if 200 ft off fencing is to be used. So first left that is considered the following figure. Now let X PD lent. And why be the bit off? Direct, angular? Mhm. It is given in the question that three sides off the field offense fence on a total fencing used this 200 ft, therefore, X plus y plus X question blunder y plus two X wells to 200. Premier, we will get the value off by which is by equals two 200 minus. Who works this equation? We will. Marcus Equation number one Now let a body area off the field therefore equals two land off the field, multiplied by a bit equals two X multiplied by right now, Using the question number one, we will put the value of fire. X Times 200 minus two x 200 x minus two X squared. Now the above expression expresses area or field as a function off lento field. Now let us represent the area function as it affects equals two Wonder Next minus two X squared Now this area function is off. The form have affects equals two X square plus BX. Let's see where equals two minus two and b is 200. And these dear Oh, so it is that quadratic function. Now we will rearrange the area function it affects equals two minus two x squared plus 200 x taking minus two Coleman X squared minus 100 ext. Now, in order to get a complete square off expression in parenthesis, we will add and subtract if this where so a affects equals toe minus two, multiplied by X square minus 100 x last 50 square minus 50 square and simplifying it minus two x squared minus under X That's 50 square they could close, plus formally declared by the square or ill faxes minus two times X minus 50 All square. Bless 5000. Now the rule of quadratic function f of X equals toe X squared plus B plus C can be re written in the form F of X equals toe a multiplied by X minus at boy square. SK graph off F is a parabola with vortex at the market. It opens, but if a greater than zero and downward if a less than zero in the function if X minus two multiplied by X minus 50 whole square plus 5000 is minus two, which is less than zero. Does the Pell parable a form will open downwards through the maximum value function? A affects will be at the Vertex that is length off field for access X minus. Coordinate over tax off. They're Ebola now from equation one bye equals two, 200 minus. Who asks here X will be 50 ft now. We will put this value in this equation. 200 minus two multiplied by 50. The value of eyes 100 ft as the dimensional field for maximum area. Yes, 50 ft and 100 ft. Thank you.

For both pens to be the same. That means we're gonna have, uh, three y values and for exercise are being used eso to find the total amount of fence we would do three, uh, four x plus three wine equals 400 feet. Offense came and then the maximized, total enclosed area. I would just be. Why? I'm sorry. The area would equal, uh, two acts. Why? And I think, uh, in this particular case, it might be easier to solve for X. Um, yeah, Muscle for X. In this case, as if I sell for acts, I'm gonna have X equals. Uh, so I'm gonna subtract three y and divide by four. So we'll have 100 minus 3/4. Why? And then it got plugged that into my area formula area equals Why r two I times 100 money is 3/4 lying. Distributing my two y is going to give me 6/4, which is three homes, negative. Three halves. Why square? Uh, plus 200 y. Okay. So, to maximize the area we needed this would give us an equation that looks like this. We need to find the Vertex, so find the Vertex will have. Wine is equal, Teoh. Negative 200 divided buying. If we multiply that about to, we're just gonna have negative three. Okay, So negative 200 development about divided about negative threes is 200 over three and 200 over three, I believe is like 67 66.6, repeating. I'm just gonna leave. It is 200 over three until the end. I'm just so we have it. Are y value would be 66.67 feet. The problem is asking for the dimensions of each coral. So that's our Y value, according to our picture. And so now we can go back to our original Formula four acts, plus three times why, which was 200 equals 400 four X equals 200. So X equals 50 50 feet. So excess 50 feet are y values 66.6 70. And again, our picture look, something like this, that would be 50 feet, 50 men, 67 ish feet. Okay, Maybe he can scrounge up an extra foot of board to make that happen. Thank you very much.


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