after-taking-a-nutrition-class-a-big-annies-mac-and-cheese-fan-decides-to-improve-the-levels-of-prot

Question

After taking a nutrition class, a big Annie’s® Mac and Cheese fan decides to improve the levels of protein and fiber in her favorite lunch by adding broccoli and canned chicken. The nutritional information for the foods referred to in this exercise are given in the table below.
a. [M] If she wants to limit her lunch to 400 calories but get 30 g of protein and 10 g of fiber, what proportions of servings of Mac and Cheese, broccoli, and chicken should she use?
b. [M] She found that there was too much broccoli in the proportions from part (a), so she decided to switch from classical Mac and Cheese to Annie’s® Whole Wheat Shells and White Cheddar. What proportions of servings of each food should she use to meet the same goals as in part (a)?

Ahmad Reda · Accepted Answer

in problem. Three. We have a person which is big fan off ends, Mac and cheese. We decided to improve the levels of protein and fiber by adding broccoli and canned chicken. Toe the Meghan cheese in the problem. We have the information regarding American the cheese, the first servant, Percle Broccoli, which is the second servant. Chicken, the third servant and shells, the fourth servant. We have here the calories, protein Ingram&#x27;s and Fiber Ingram&#x27;s. In every servant we have here 217. 10 to 51 5.4, 5.2, 70 15 0. And finally you have four shells 260 lying and fight for this problem. We want to toe for party to use broccoli and can chicken added to mac and cheese to produce a launch that has 400 colorless and 30 g off proteins and 10 g off five. We can do this by mixing the third the 1st 3rd servants. The first servant will be X one multiplied boy. The first nutrition information of the first servant 270 10, and to plus the amount of the second servant multiply by its information. 51 5.4 5.2 plus exit three. Well, the boy boy 70 15 and zero. Then using any metrics program to solve this system, we can write the augmented metrics first augmented metrics to contains thes three victors. Besides this vector 270 10 to 51 5.45 point two, 17, 15, 0, 400 30 and 10. Using software, we can get it. The reduced patron form in that in the reduced its reform, we have one zero zero. And in this month we have Oh, boy, nine it 58 We have zero one 1.5 for 39 and we have here 00 one 4.7 It 70 And using back substitution. Starting from the third equation, you can get that exit three equals 4.787 Next to equals, 1.5 for 39 and x one equals almost one. And we can approximate the amounts. As for Mac and cheese, we use one making the cheese. We use one servant for broccoli. We use 1.5 servants and for candid chicken news. Oh, point it. Servant for both v off the problem. We were replaced Mecca and the cheese with shells. Not to use too much broccoli. Then we will have the same augmented metrics in the above need. Except for the first column will be repressed Boy. The fourth column. 260. 260 mine five. Then we have the second two columns as it is 51 boy 55.45 point two, 70 15 0. And the right hand. So it as it is, you want to get 400 calories that the grams of protein and 10 g of fight using any metrics software, you will get this metrics as a Nick Wavell int toe. One 00 one point one zero 10 four point plane zero zero one on. We have here one which stays. We will. He will use 1.1 Servant off shells. 0, 00.9. Servant off broccoli. Finally, one servant off candy check. And this is the final answer of work to be. And this is the final answer off body

Ankit Gupta · Answer

let the servings off roast beef Be excellent, sir, on the serving So big potato, the number of servings of being protected Levi and the servings off broccoli. Be said Graham. No. Given conditions in our question can be translated as 300 plus as we get 300 calories from roast beef 100 calories from big potatoes on 50 calories from Rome. Roast being 50 calories from broccoli so the total calories intake is 800. Which weekend simply fires six x plus two y equals plus, then equal 16 named, he says. Our first aggression it&#x27;s now. Secondly, we are given that Tortola protein in day is drunk. The X Plus five i plus five said, equals 55. So simplifying this equation we get four x plus y, Blessed said, equals 11. No third Levy are given that dont buy plus hundreds of equals 20. Therefore, we can simplify this equation. Also fit together, which we get went by. Plus five equals 11. Need the seller toward a question now multiplying equation too. You question, too is here multiplying whole aggression to buy fine and subtract three from it. So we get 20 x plus five by plus fights that equal 55 subject immigration trees zero X place by plus five Said equals 11. Subjecting this equation, we can change December&#x27;s symbols off this equation and righteousness. 20 eggs plus four way the skins is out is 44. Therefore, our equation is simply thank this equation we get five x plus y equals 11. So named this as our fourth equation now subjecting aggression number food now, firstly, multiplying aggression with No, it&#x27;s using a question one on the question toe to eliminate said We will, right six x, which is our first equation six x plus two y plus n equals 16 on our second aggression. Is this so Linus subjecting second equation from this kid? So we have eliminated here the sector Look at you. X place by equals five. It is a 50 question. So we have eliminated said from both these equations. Now subtracting equation five from a question number four bigot five x plus y equals 11. Is that a question? Number four and two x plus y equals five is a question Number five you get three. X ik was six. So we have found the value of exited six by three. That is to now substitute this value of X immigration number five. So we get doing toe plus by equals five. Therefore, four plus y equals five so we can subtract four from both sides and get five minus y equals five minus four. That is one. Now we&#x27;re good, Good, the value off X and y. Now, to find the value of said, we&#x27;ve been substitute this value of via immigration number three, so we can write it. As in this equation, one bless fights n equals 11. So the value of Zaid is then subtracting one from both sides, and thus that becomes 10 by five. That is to therefore, we have got that the number off calories in pigs. So the number off servings off each of these roast beef, big potato and broccoli should be do one and do that is two servings off roast beef. One. Serving off big potato on to serve servings of broccoli again. So this is a required solution

Matthew Dunham · Answer

okay for this problem, We&#x27;re gonna be writing a system of inequalities and then graphing it to find the feasible solutions. We have food, M and food. End food. M is our X variable Food ends are y variable, and we have three different nutrients. Um, we&#x27;ve got calcium, iron and vitamin A that with medium with. So it&#x27;s three different equation. So let&#x27;s start with calcium. We&#x27;ve got 30 units of calcium for food A and we&#x27;ve got 10 units. Sorry, not food. A for food M and then 10 units for food end. And then we&#x27;re told that there is a minimum requirement in the diets. That means we have to do greater than or equal and 3 60 Then we move on to iron and we&#x27;ve got 10 for each. So 10 X plus 10 y and the minimum there is 160 and then we have to do vitamin A, which has 10 for food em and 30 for food end. And that minimum is to 40 and then we have X is greater than or equal to zero. And why is greater than or equal to zero? Because you can&#x27;t have negative, um, ounces of food. OK, so that&#x27;s our system of linear inequalities. Here is the screenshot from Dismas. When I plot those as equations again, I applaud him as equations on Lee because I don&#x27;t want this to be really hard to read with all the shading. But remember, these are all greater than or equal to on the actual problem. And so you see the graphs because we&#x27;re talking about greater than or equal to. We have to shade above each of these lines. So it&#x27;s gonna be the region is gonna be in the first quadrant. It&#x27;s going everything above this line here and then everything above the green line right there or the blue line, I should say, And then the green line down here. And then we just do a big shading everything up here. This is the region of solutions of feasible solutions. Um, and it is unbounded. Okay, so there is No, there&#x27;s no maximum. I mean, you can you can have uses many ounces. Um, there is no maximum situation. So there&#x27;s a There&#x27;s your shaded region. You have your corner points 0 36 What that means is that you could have zero of food M and, um, 36 of food end, and then over here, you could have 24 of em and zero event. Now, when you talk about a hospital, you&#x27;re probably they&#x27;re probably wanting to use both foods. More than likely they&#x27;re gonna use something in inside that region here or at these points instead of at the max at the extremes. But anyways, there&#x27;s a shaded region, and that is it.

Ankit Gupta · Answer

indiscretion. We have to find out the number off servers off each food item required to get the desired nutritional values reassigned. Available extra beef by two Baked potato on They do run carbo strawberries. So now we&#x27;re going to put all this information and a tableau for with the columns calorie carbohydrates, gal cheer for us. We know down there required on, uh, essentially you to use that for 85 calories, 41. Put five guards and terrify patchy. We don&#x27;t the rest of the information as given in the question. Call them by Colin now, before creations out of them that to 45 x plus 1 45 45 people sport 85. That&#x27;s the first bullet for calories. Then we formed the creation for the second quarter. That study full wife has tens at equals. 41.5. Afforded her column. That&#x27;s for culture. That&#x27;s nine X plus eight by anyone people in the study. Now we saw this using the P I. A T forecast Lido. Now notice carefully how we are going to use David. That&#x27;s a matrix form. In order to solve this, we go to the major exception. We headed off 368 Make it at the and Before Matrix and our values in the room so that it was three present Our regressions that&#x27;s do 40 by rates less 1 45 by blessed 45 said That equals 45. Then we end our secondly creation, that&#x27;s, you know, weeks. Plus, we said 34 by Andre Said speak with 41.5 that in the last few creation that Nynex it. Why, Anthony One said that equal title fight. Now we&#x27;re gonna go to the Matrix option. The are That&#x27;s a reduced Roy Kolya before we get to know more about it later. And our matrix okay in the bracket in the button. In the last column, we get our values off X. Lyons said. X is not going to first. Why is running at this point? Therefore, final answer. We know that X because what we do fire Why once, when that&#x27;s a sequence 0.75

Taylor Shimono · Answer

So for this problem, in part A, we want to write a system of inequalities that are going to tell us the different combinations of amounts of food, X and food. Why that we can eat in order to meet the daily minimum of each essential element in our diet, Um, for one person per day. So I want to have in turn all the information from the word problem into a table. Uh, and I did it this way because I find it convenient to be able to organize my information like this. It makes it a little bit easier for me to see what&#x27;s going on and hopefully try to put it together in some meaningful way. So since we have three, uh, requirements on calcium, iron and vitamin B, All right, that means that I&#x27;m going to be using three equations to represent, uh, these thes essential elements that we need. So I&#x27;ll go and start by writing, 01 inequality for our amount of calcium. So we know that each unit of food exit we eat is going to give us 20 units of calcium. So I&#x27;m going to well, 5 20 by the number of pieces of food XO eat and we&#x27;re going to add this to 10 times the number of pieces of food. Why that we eat. And we know that the amount of calcium that we get from this food has to be greater than or equal to our minimum of 300 units of calcium. And we can go and do the same thing for each of our other two elements. Um, so next for iron, we know that we need ah 15. We&#x27;re going to gain 15 units of iron A per unit of Food X, that we&#x27;re going to multiply 15 my ex and we add this to our total iron is going to come from our 10 units of iron that we get per piece of food. Why? And this has to be greater than or equal to our daily minimum of 150 units of iron. And finally, for vitamin B, we get 10 units of Ivan be per unit of X plus 20 units of vitamin B for food wife, and this has to be greater than or equal to our minimum of 200 units, vitamin B. And lastly, we&#x27;re just going to set some restrictions on our amounts of Food X and whether we eat almost. We know that we have to eat a real number of food, so X has to be greater than or equal to zero. And why must also be greater than or equal to zero? And that&#x27;s just because the fact that we don&#x27;t know what it means to eat in negative amount of food. So for a party we have or five inequalities in our system here, next for Part B, we want to grab each of our inequalities, so I&#x27;ll go and start the bottom when we know the X must be greater than equal to zero so we can just draw our line. But X is equal to zero, and we also know that why must be greater than or equal to zero. Um, and then for our first equation, I&#x27;m going to rewrite this in slope intercept form because I find it a little bit easier to Graff. So first I&#x27;m going to subtract 20 x from both sides of my equation and then finally divide by scent and we get wise greater than or equal to 30 minus two x, so that means that I&#x27;m just going to put my why interception or 30. And we have a slope of negative too. I&#x27;m so forever to units, you move downward, we&#x27;re going to move one unit to the right. And so when we plot a few extra points on a graph, our equation overseas, we are inequality. Oh, for calcium is gonna look like this. And through that, why must be greater than this value? We know that we&#x27;re going to be shading the region that&#x27;s going to be up upwards of our graph. So to the right in this direction, um, for our next graph again, I&#x27;m going to rearrange to being slope intercept form. So we get 10 wise greater than or equal to 1 15 amount of 15 X and again dividing by 10. We get wise greater than equal to 15 minus three have x. So again, we know what are Y intercept is, and we have our soap. So we are going to be able to pull out a few more points on a graph and connect them once again. Um, and then again, this time we have why is greater than 15 minus three or two X so that tells us that we&#x27;re going to again be shaving above this region here and our third inequality. When we rearrange, we get 20 wise greater than or equal to 200 minus 10 IX and dividing by 20. We get y is greater than or equal to 10 a minus 1/2 X. So once again, we&#x27;re going to be able to pull out a few points on a graph and connect them, and we see that we&#x27;re going to be shading of both. So once we have a lot of our inequalities plotted, we can go ahead and figure out that the region that&#x27;s going to be encompassed by all of them is just going to be in the first quadrant above each of these three inequalities. And once we finish the shading, we have a completed autograph here and lastly for apart see of our problem. We want to pick two solutions within our solution set and determine what they mean in the context of the problem so I can see that the a point that&#x27;s going to lie right around here at 14 10 is going to be on the graph, so it&#x27;ll been picked that as one my points. And so we know that this is in the four ex wife. So essentially, what the 0.14 10 is telling us is that if we are to eat 14 units of food X in 10 units of food, why that&#x27;s going to give us enough of our nutrients to meet our daily minimum in another point, that we can go and pick is going to be somewhere right around here along the y axis. So, for instance, D 0.35 zero or 0 35 let&#x27;s cross this out here 0 35 And what that&#x27;s telling us is that if we don&#x27;t want to eat any of food, ex say we&#x27;re allergic to it. You need to eat 35 units food white in order to make up for our lack of a few necks and still get all of the A minimum minimum requirements that we need for our nutrients.

Christine Anacker · Answer

it was holding. The fishing problems are equation, so we know we need or we know that we&#x27;ve got 23 grams of protein report grams of corn, nine grams protein milk. I need 47 grams protein total. He&#x27;s got two grams of carbs in the court grounds, of course. Corn, grams of carbs. A little less when we have nine grams. That&#x27;s right. We&#x27;ve got 10 milligrams of calcium in the port and middle ground Worry and 10 300 milligram. Um, we need a total of six. Now. The other one&#x27;s Aaron grounds one milligram, if you want to. Doesn&#x27;t matter. I go there all this immunity. All right? Let&#x27;s Yeah. Oh, have there. All right, I&#x27;m gonna my third row. She&#x27;s gonna change it in grams in weight. You 11 30. Awesome. I take my my first real time. What? You know, when you No, my When, what? Seven minus one. Oh, 11 Um um Well, it&#x27;s not great. No waas over 20 one. Well, something. My 13 waas over one one. What? Look, you What do you A day of work or 100? You? I want one. All right, so we&#x27;re going wound about it by one? No. Well, but when it&#x27;s getting one, you know about about All right. Oh, we&#x27;re gonna do a double step here. Just waiting those numbers below. Well, would work. All right, little one, over. Warren, I would like one. What? No, you really? If you want. Or just remember that are, for one. What? It was not one for.

Problem

The Cambridge Diet supplies .8 g of calcium per d…

Problem 3 Easy Difficulty

Answer

(a) 1.0 serving of Mac, 1.5 servings of Broccoli, 0.8 servings of Chicken
(b) 1.1 serving of Shells, 0.9 servings of Broccoli, 1.0 servings of Chicken

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Catherine R.

Caleb E.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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(a) 1.0 serving of Mac, 1.5 servings of Broccoli, 0.8 servings of Chicken(b) 1.1 serving of Shells, 0.9 servings of Broccoli, 1.0 servings of Chicken

Linear Algebra and Its Applications

Grace H.

Catherine R.

Caleb E.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Caleb E.

Michael J.

(a) 1.0 serving of Mac, 1.5 servings of Broccoli, 0.8 servings of Chicken
(b) 1.1 serving of Shells, 0.9 servings of Broccoli, 1.0 servings of Chicken

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications