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A farmer finds that if she plants 55 trees per acre, each tree will yield 55 bushels of fruit. She estimates that for each additional tree planted per acre, the yie...

Question

A farmer finds that if she plants 55 trees per acre, each tree will yield 55 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels How many trees should she plant per acre to maximize her harvest?treesGive your answer rounded to the nearest whole number_

A farmer finds that if she plants 55 trees per acre, each tree will yield 55 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels How many trees should she plant per acre to maximize her harvest? trees Give your answer rounded to the nearest whole number_



Answers

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Yeah. Mhm. Yeah. Yeah. Yeah. Yeah. Yeah. Mhm. Yeah. Yeah. Yeah. Mhm. Okay. Mhm. Yeah. Mhm. Right. Yeah. Yeah. Mhm. Yes. Yeah. What's your take? Mhm. Yeah. Mhm mm. Yeah. Mhm. All right. Yeah. Yeah. Yeah. Yeah. Okay. Uh huh. Yeah. Mhm. Yeah. Yeah. Mhm. Yeah. Mhm mm.

We make a little table of what we have here. We know that this is the number of trees I'm gonna call that X, and that's going to be the number of trees for Acre. And we know that if she plants 75 trees in an acre, that let's call this why we know that the number of bushels that will be produced by a tree per uh per tree is equal to 20. And then it tells us that if she increases the number of trees by one per acre, that shall decrease the harvest or the number of bushels that they'll have. And you know, if you increase this, this is going to go down to 14 now. What's the harvest? Because we want to maximize the harvest and the harvest is the product of X Times y that if we take the number of trees and this is not a given acre, 75 trees per acre times 20 barrels per acre, that's going to be the harvest 76 times 17. That's going to be the harvest per acre and so on. Well, we need a generalization for this. And so let's let X stand for again this tree. And now we want to find out what this is in terms of X. And it's a linear relationship because as X goes up by one, why goes down by three? So we know that the equation between X and Y there's a slope of negative three. That's the drop per X. We just need to figure out what that Y intercept is. So if we plug 20 in place of why and 75 in place of X and then add this product to both sides, we end up finding out B is equal to 245. So if you know how many acres, how many trees are going to plant per acre, you'll know that negative three X plus 2 45. That's how many. Uh, that's how many bushels you would produce per trees. So this is negative. Three X plus 2 45. Now how do we find the harvest? We take X and we multiply it by negative three x plus 2 45 the same way we did the other ones. So we know what the harvest is as a function of X, the number of trees per acre, and it's X times this factor. Now we want to find what the maximum harvest is, and we can do this by looking at what the intercepts are for this quadratic, and we can see that the quadratic would intersect at an X value of zero effect zero harvested zero. We know that it's a quadratic. I mean, I can distribute it out like this with No, it's a quadratic that opens downward because that lead coefficient is negative and the other intercept is going to take place when negative three X plus 2 45 is equal to zero. That's when negative three X is equal to negative 2 45. And if we divide by three, Unfortunately, I can see that that is not going to go into their evenly. And it comes out to be that X is 81 2 3rd. So there's an intercept over here at 81 2 3rd, which means that the middle this parabola is going to do this and come downward. And we need to know what this is, which is going to be halfway between these two. So if I divide that by two, I find out that at 40 and one excuse me. If I divide that, we end up getting 40.83 repeating. Which means that approximately at 41 if we have the number of trees being 41 let's figure out what the harvest is for that and plugging 41 back in here because we can't have a fraction of a tree. And so if I take 41 times and then take negative three times 41 plus 2 45 we find out that that would give us a harvest of 5002 bushels on a certain acre. And look what happens if we go to 40 if we go less and that would be taking 40 times negative three times 40 plus 2 45 and that is only 5000 bushels. And if we go higher, we're going to end up being lower. This will be the best combination that if we have 41 trees, 41 trees per acre, that she would yield 5000 two bushel in an acre

In this question, we have to find how many trees but a great should be planted for much money after Jean has given us for DPU. Marjorie, a number of trees are 20 threes for ego. The maximum yield in this case is 40 bushes upon trees died 20 trees upon acre which gives the in as 800 bushels will ego. So you have to go on every bushels for ago. For the no, that's gonna be 40 minus and 20 plus. And because for every tree drone you decreases by one. Next, you're going to maximize this function, which is going by my old friend. Religion equals do minus ends for just ready and less 800. So do Max the number of trees that need to be planted. We find the X coordinate off the board. Eggs here is minus one B s trendy and see is 800 on max and that is the trees that need to be planted for eager and minus b upon to it. That's the export text, which is 10 trees for eagle as calculated on your screen


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