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An noints crtn credi| Iencinz oppusite lenccd in Lsine Kind rectangular plot of land rcmaining IWO sldcs Yi usc heavy-duty lencie that costs S6 per foot; #hile the ...

Question

An noints crtn credi| Iencinz oppusite lenccd in Lsine Kind rectangular plot of land rcmaining IWO sldcs Yi usc heavy-duty lencie that costs S6 per foot; #hile the 5ides Whal are thc dimensions will uSc standard fencing that costs $4 per foor. Mcco 0 # COSl 0f81,9202 meclangulun plot of greatest aneninJecg

An noints crtn credi| Iencinz oppusite lenccd in Lsine Kind rectangular plot of land rcmaining IWO sldcs Yi usc heavy-duty lencie that costs S6 per foot; #hile the 5ides Whal are thc dimensions will uSc standard fencing that costs $4 per foor. Mcco 0 # COSl 0f81,9202 meclangulun plot of greatest aneninJecg



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decide which of the following properties apply to each function. (More than one property may apply to a function.) A. The function is increasing for $-\infty<x<\infty$. B. The function is decreasing for $-\infty<x<\infty$ C. The function has a turning point. D. The function is one-to-one. E. The graph has an asymptote. F. The function is a polynomial function. G. The domain of the function is $(-\infty, \infty)$ H. The range of the function is $(-\infty, \infty)$.
$y=e^{x}$

So we're gonna make a rectangle and we're going to do it with two sides. One of the sides will call X, and one of the sides will call W and the science that are X cost $8 per yard and the W's cost $12 per yard. And our total budget is $4800. So the cost is going to be for that rectangle. The perimeter is two exes and two w. So we're gonna have to times x times $8 per yard. Plus, we're gonna have to w so two times W times, $12 a yard. We know that the area we'll call it the function A of X is going to equal one of those sides times the other side. And we want to maximize the area, right? We want the dimensions for that. So we want this function to be only in one variable. So we're gonna come over here and we're going to solve our cost equation for W. So we're gonna have 24 w two temporal was 24 is equal to 4800 minus two times eight is 16 so 16 x. And that tells us w equals 4800 over 24 minus 16/24 x So w equals 2024 goes into 4800 200 times 200 minus 16/24 is the same thing as 2/3. So 2/3 X. Now we can substitute the W into our area formula. So we're gonna have X times 200 minus 2/3 acts. And if we distribute the X, we see quadratic emerging. So 200 X minus 2/3 X squared. And we really should put this in general form. So I'm gonna do a little whiteboard magic and swap the order because we want the X squared term to be first. That way we won't accidentally pick up the wrong coefficients. So this is our area function, and it is a parabola and furthermore, this leading coefficient. Sometimes we call that a is negative, which means our parabola is opening downward. And that means right there at the Vertex we have a maximum so we can get our X for the maximum by finding the x co ordinate of the Vertex. So remember our coefficients, we call them A and B and our Vertex X coordinate. The whatever the independent variable coordinate of the Vertex equals negative B over to a. So it's gonna equal to negative 200 over two times negative 2/3. And that equals, Let's say we got negative 200 over negative for his positive 50 times three is 150. So that's one of our dimensions. So our area will be 100 and 50 yards by, but we right by So we don't confused that with the letter X might be a better way to do this. So say, but didn't that come off there? We go to 150 yards. By now. We have to find our other one. It's gonna come from W W. Is equal to 200 minus 2/3 times acts. So 2/3 times 150 Well, with 1/3 of 100 fifties 52 3rd is ah, hundreds. So 200 minus 100 equals 100. So that's our other dimension. So our max area is going to be 150 yards by 100 yards

Probable question. About 35 in this case were given a rectangular lee region, which were supposed to fence. But the cost of fencing is different for the different sites. So if the's states that X and we're here, this is why and the worst of X, which is East west. The fosters for extra cost, a stray dollars butter. Now the unit is a yard, and for that would survive the cost of speed dollars for yard. We're supposed to find the dimensions of the largest possible area, given that the cost off the total cost of the fences 4808 $100 so total cost of the fence would be explosive exchange well plus five. Plus why times hit. This is already given US 4800 so this will be 24 x plus 16. Why? Let's divide us doing before So we have 100 or do hate X plus. This will be four X over here. It should be okay, So can we divide? This father looks divided further by. We're here in the sixth. In theory, waiting about Ford's four times six is 24. Let's go right back to West. So we have 600 0 to 3 x plus. This is why over here we have three experts to why From here we can write the value off while a 600 minus three x Otherwise, we can also write the value via 600 minus three X over. Now, based on this condition, we have maximized the idea the area will be exchange while so let's obstruct the value of why. Here we have 600 minus three x or let's above the bracket. So we have 600 X minus three X square working now this is a quadratic equation which is opening a parable of Jews which is opening downwards so its maximum value occurs at the cortex which is managed people away B In this case, the 600 were too. On day, two times A in this case, is ministry over to So here we get 600 or six, which is nothing but 100. So the value of X comes out as 100 yards on. Then we talk about the wire by 600 ministry X over two so by 600 minus three times X which is 100 over to. So this is 300 over to which is nothing but 1 50 yards. So these are the dimensions off the rectangle of feel such about the cost US 4800 on the years maximum.

In this problem. We have you in the dining since off the tangle to be a and B. Where is the Green Train and Bista? Red site for the Redford. We have the cost of fencing us $2 per feet. And for the green side, we have $2 per feet. So we have been given that the total cost or fencing? Yes, $6000. So we have free multiplied way Do a plus two multiplied play Do be has six tell isn't so we can say that B equals 3000 minus three. A. By do now we know that the area off a rectangle is given by the product off its length and expect which in this case, this e n b how from the every question we can write 3000 minus three A by do which gives us the idea Function s 3000. But I do e my house three by two a square No, to find out the critical point, that is to for a year to be maximum be differentiated with respect to a So we have you Dash s 1500 minus six by two a And then he quitted to theater to find the critical part. So we have a s 500 ft. So now that we have a you can find out, be easily from the about immigration. So we have 3000 minus three, multiplied by 500 divided by do which after evaluating, gives us be as 700 from D feet.

I think I got all the details here. We have the farmer's barn and on the north wall he's gonna build a pastor or a field or whatever. Uh huh. Okay. And he doesn't have to put fence on the where the barn is. Okay. And then on one fence his neighbor is going to pay for part of that. I guess I should put that at the end of it barn. Maybe his barn is right on the edge of the property or something that. Okay so that's the blue one, his neighbor is going to pay for half of that. And then um The Green one, he has to pay all of it. And so what and the most he will spend is $5,000. And the fence costs $20 per foot. And so we want to maximize the area of the um pasture field or I forgot what it was. So we'll just call it area. So we want to maximize area equals. Well what's the equation for the area of a rectangle? Well it's like times with but we need to give our specific names, let's call this one eggs because it's horizontal. And then these two will be linked Y. So that's the length. So we're going to maximize X times Y. Subject to, okay, the total cost has to be less than 5000. So if it costs $20 per foot and he's paying for all of it and he has to pay for Y. Feet. That's 20 Y. Plus, he has to pay for exhibit 20 x. Plus, he only has to pay for half of this. So instead of being $20 per square foot, it will be $10 per square foot and that's why. Okay, I can't stand that. Why I got to fix it. Okay, 20 Why? So the first one is the cost for this part. Right here. 2nd 1 20 X costs for this third one. His cost for this His neighbors have to pay the rest and we need that to be less than or equal to 5000. So let's just say equal to. All right. So, if I saw if I simplify that a little bit is 30 Y plus 20 X equals 5000. All right. Any time to find a maximum. You're gonna take the derivative of the thing and then set it equal to zero. But the thing has too many variables. That's what the constraint is for. This is the constraint restrains the size of things depending on how much they cost. Okay, So, we need to take out either X or Y. So you can decide you want to solve this for X or Y. I don't think it's gonna matter. Let's just solve it for why 30 Y. Is 5000 minus 20 X. Yeah, I would have got 2000. If I would have solved drags, I wouldn't have fractions on both of these because the zeros cancel and we get 503 -2/3 x. So area is x times y. Which is 503 -2/3 x. Which is 503 X -2/3 x squared. All right now we're gonna take the derivative with respect to X. So we'll call it a prime derivative of a constant times X is the constant minus the constant times the derivative of X squared which is two X. We're gonna set that equal to zero. We're going to multiply this whole equation by three. So there are no fractions 500 -4 x equals zero. So for x equals 500 X equals 500 over four, which is 1 25. And then remember it wants to know the dimension. So it wants to know why also and it was 503 minus two thirds X. Is that right? Minus two thirds X. So that's 503 -2/3 times 1 25. That's 500 -2/53. So 253 Or uh 83 and 3rd. So he should make it um North wall, North fence, sorry for what we're talking about. North fence, ft. And then the west and east vince 83 and a third feet each


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