Question

These questions come from this Linear Algebra book (picture attached ). I can not understand how to solve the questions 3; so please help me explain how you can solve it (as much as you can). Please do it carefully. Many thanks to you.

Linear Algebra Diet Problems Text Reference: Section 1.10, p. 93 The purpose of this set of exercises is to provide examples of vector equations which result from balancing nutrients in a diet. Section 1.10 shows how use a vector equation a+ag..+a,-b to model a diet with a specified nutritional intake. Each vector a, lists the nutrient composition. of one unit (usually 100 grams) of foodstuff, and the corresponding weight , is the variable that represents the amount (number of units) of that foodstuff to be used in the diet. The vector b lists the amount of each nutrient that must be in the diet. Table 2 below is a listing of the nutritional value of many foods found in a typical kitchen. The nutrients are given per 100 grams of foodstuff. This data is taken from the USDA Nutrient Database for Standard Reference available at the United States Department of Agriculture web- site (http://www.nal.usda.gov/fnic/ foodcomp). The columns represent respectively the following foodstuffs: beef, brussels sprouts, carrots, chicken soup, egg, feta cheese, grape- fruit, lentils, lettuce, milk, mushrooms, oil, onion, rice, salad dressing, salmon, soy sauce, spinach, tomato, and vanilla ice cream. Table 1 gives the standard serving size for each of these foodstuffs, and also gives a key to the columns in Table 2 Questions 1. Low carbohydrate diets are popular for weight loss. Compute (by hand) the amount of carbohydrates in each of the following dishes, and determine which would be better for such a dieter to choose. You will first need to use Table 1 to convert the kitchen measures into 100 gram units, then use Table 2 to find the amount of carbohydrates in each ingredient. Spinach omelet: 1/4 cup spinach, 2 eggs, 1/8 cup milk, 1/2 Tbsp. oil Greek salad: 1 cup lettuce, 1/4 cup feta cheese, 1/2 of a tomato, 1/8 cup salad dressing 2. To make a stir fry, fry beef and onions in a wok with oil, and top it with soy saud auce. a) You would like to make a stir fry with a total of 6 g carbohydrates, 50 g protein, and 3.5 mg vitamin C, and you'd like this dish to contain only 575 calories. Use Table 2 to set up a matrix equation which could be used to determine whether it is possible to make such a stir fry. Describe the steps you take to produce the vectors in the equation. b) Find a precise recipe for the stir fry in part a). Convert your amounts to common kitchen measures using Table 1. 3. Table 2 has been incorporated into the matrix A which accompanies this exercise set. What does the jth column in this matrix represent? Which entry in this matrix tells you how much vitamin C is found in 100 g of vanilla ice cream? 4. A particularly math-savvy sumo wrestler wanted to adhere to a strict diet to maintain his weight and strength. Table 3 lists his desired nutritional intake for one day. The entries in Table 3 are stored in the vector vi which accompanies this exercise set. Using Table 2 he was able to decide on an optimal diet to give him this combination of nutrients. How much of each of the above foods were in his diet? 5. The United States Food and Drug Administration (FDA) provides Recommended Daily Val- ues for use on food labels. Table 4 gives the FDA's recommendations, which are also stored in the vector v2 which accompanies this exercsise set. Is it possible to combine the foods from the table to approximate these nutritional values? Serving Size 6 oz 170g 1/2 cup-78g 1 carrot 61g 1 cup 240g 1 egg=6lg 1/4 cup-38g 1/2 fruit 123g 1 cup-198g 1/2 cup-28g 1 cup 244g 1/2 cup 35g 1 Tbspa 13.5g 1 onion-110 1 cup-158g 1 cup-250g 1/2 fillet 124g 1 Tbsp-18g 1 cup 180g 1 tomato 123g 1/2 cup 66g Key Number Foodstuff Beef 1 Brussels Sprouts 2 3 Carrots Chicken Soup Egg Feta Cheese 7 Grapefruit 4 6 Lentils Lettuce Milk 10 11 Mushrooms 12 Oil Onion 13 14 Rice Salad Dressing 15 16 Salmon Soy Sauce 18 Spinach 17 19 Tomato Vanilla Ice Cream 20 Table 1: Serving Sizes of Various Foodstuffs Foodstuff (see key for names) 6 Nutrient (units) 10 8 5 152 10.33 11.44 1.04 492.5 2 3 4 116 61.44 3.29 3.34 Calories (kcal) Protein (g) Fat (g) Carbohydrates (g) Calcium (mg) Iron (mg) Magnesium (mg) Phosphorus (mg) Potassium (mg) Sodium (mg) Zinc (mg) Copper (mcg) Vitamin C (mg) Thiamine (mg) Riboflavin (mg) Niacin (mg) Pantothenic Acid (mg) Vitamin B6 (mg) Vitamin B12 (mcg) 263 14 73 5.3 2.5 30 55 215 39 43 1.62 26 9.02 14.2 2.55 1.03 38 21.3 11.5 51 .19 2.37 4.66 4.09 7.68 11 20.14 8.6 10.1 7.1 119.4 .05 13.44 45 19 36 7 36 1.2 27 10 42 3.1 27 12 3.33 1. 1.19 .65 6 19.2 15 36 6 20 93.4 151.5 337 180 211 56 317 44 30 148 129 323 45 354 369 290 367 101 61.8 270 8 49 1116 2,88 69 21 35 25 .07 38 ,01 5290 J43 33 1.27 251 92 .013 4 037 .032 044 083 .047 38.1 .034 94 1.5 24 62 .107 .08 0 154 844 9.3 .038 .169 .11 25 097 ,059 ,03 .07 .044 .162 084 5 314 399 .073 1.06 .638 .02 191 4.63 .607 928 1.8 .058 991 283 252 4 2.27 34 ,17 967 424 197 15 934 .042 178 047 178 147 02 13 509 109 042 357 1.69 2600 Vitamin A (IU) 719 126 28129 447 259 654 Nutrient (units) Foodstuff (sce key for names) 16 15 14 17 18 12 19 20 11 13 Calories (kcal) 38 130 448,8 149 60 23 21 201 884 3.5 2.09 1.16 2,69 0 25.56 10.51 2,9 85 Protein (g) 0 28 50.1 4.42 J 26 33 Fat (g) Carbohydrates (g) Calcium (mg) Iron (mg) Magnesium (mg) Phosphorus (mg) Potassium (mg) Sodium (mg) Zinc (mg) Copper (mcg) Vitamin C (mg) Thiamine (mg) Riboflavin (mg) Niacin (mg) Pantothenic Acid (mg) Vitamin B6 (mg) Vitamin B12 (mcg) Vitamin A (IU) 100 16 11 4.64 4.65 23.6 8.63 28.17 2.5 5.57 3.75 0 136 .18 20 10 0 17 20 128 22 2.38 1.24 38 1.2 .99 3.57 45 ,09 01 10 10 12 33 40 87 11 14 104 1.22 33 43 295 130 56 24 105 0 157 414 212 35 370 7.5 466 222 199 .04 86 5586 70 4 80 06 49 73 19 71 43 76 09 .69 069 135 492 0 .06 099 .174 074 023 3.5 0 6.4 0 9.8 19.1 102 163 013 .196 042 0 .059 152 3.951 095 059 .041 02 148 449 0 073 236 .048 24 49 1.476 0 8.526 40116 .628 .116 2.2 247 .106 39 0 865 376 145 581 .116 .093 .097 231 2 242 .08 .048 3.46 0 0 0 39 0 0 136 0 8190 623 409 Table 2: Nutritional Values of Various Foods per 100g of foodstuff rle le le se le olelel le on-oo en 2235432 Nutrient Amount 8279.12 kcal Calories Protein Fat Carbohydrates Calcium Iron Magnesium Phosphorus Potassium Sodium Zinc Copper Vitamin C 608.81 387.60 604.48 4067.42 mg 93.34 mg 1714.73 mg 8488.03 mg 18023.48 mg 8846 38 36009.75 mg mg 6.67 mcg 604.06 mg 6.77 Thiamine mg Riboflavin Niacin 10.61 mg 28212.10 mg 103.11 Pantothenic Acid Vitamin B6 Viamin B12 Vitamin A mg 189.81 mg 51.78 meg JU 95382.93 Table 3: Sumo Wrestler Diet Nutrient Amount kcal Calories 2000 Protein Fat Carbohydrates Calcium 50 g 65 300 g 1000 mg 18 Iron mg Magnesium Phosphorus Potassium Sodium Zinc Copper Vitamin C 400 mg 1000 mg 3500 mg 2400 15 2000 mg mg meg 60 mg 2 Thiamine mg 2 Riboflavin mg 20 Niacin mg 10 Pantothenic Acid Vitamin B6 mg 2 mg Viamin B12 6 meg IU Vitamin A 5000 Table 4: FDA Recommended Daily Values

Answer 1

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