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 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 lists 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 website, www.ars.usda.gov. The columns represent respectively the following foodstuffs: beef, brussels sprouts, carrots, chicken soup, egg, feta cheese, grapefruit, 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. Key Number 1 2 3 456 xa, +xa+...+xa, b 7 8 9 10 11 12 13 14 15 16 17 18 Foodstuff Beef Brussels Sprouts Carrots Chicken Soup Egg Feta Cheese Grapefruit Lentils Lettuce Milk Mushrooms Oil Onion Rice Salad Dressing Salmon Soy Sauce Spinach Serving Size 6 oz. = 170 g 1/2 cup = 78 g 1 carrot 61 g 1 cup = 1 egg 1/4 cup 1/2 fruit = 240 g 61 g 1 cup = 1/2 cup 1 cup = 1/2 cup = 1 Tbsp. = 1 onion = = 38 g 123 g 198 g 28 g 244 g 35 g 13.5 g 110 g 1 cup 158 g 250 g 1 cup 1/2 fillet 124 g = 1 Tbsp. = 18 g 1 cup = 180 g Nutrient Calories kcal Protein g Fat g Carbohydrates g Calcium mg Iron mg Magnesium mg Phosphorus mg Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley Potassium mg Sodium mg Zinc mg Copper mcg Vitamin C mg Thiamine mg 1 Tbsp. 18 g 1 cup = 180 g 1 tomato 123 g 1/2 cup = 66 g Table 1: Serving Sizes of Various Foodstuffs Application Project: Diet Problems Riboflavin mg Niacin mg 17 18 19 20 Soy Sauce Spinach Tomato Vanilla Ice Cream Pantothenic Acid mg Vitamin B6 mg Vitamin B12 mcg Vitamin A IU 2 3 39 43 2.55 1.03 0.51 0.19 8.6 10.1 1 5 6 7 8 9 14 215 152 263 30 116 10 61.44 3.29 26 10.33 14.2 9.02 1.62 11.5 2.5 0.55 11.44 21.3 0.1 1.04 4.09 7.68 0.38 0.2 3.34 0 7.1 20.14 2.37 4.66 7 36 27 10 119.4 36 1.1 3.1 1.2 0.5 0.6 0.05 27 20 15 211 56 13.44 93.4 151.5 367 69 49 0.38 0.01 42 492.5 11 19 1.19 0.65 0.12 3.33 4 9 19.2 8 36 6 44 30 148 337 9 180 45 317 323 45 101 61.8 129 369 290 21 35 354 270 1116 0 2 8 5290 0.33 0.2 0.4 0.92 2.88 0.07 1.27 0.25 0.143 0.083 0.047 0.1 0.013 0.032 0.044 0.251 0.037 0 62 9.3 0 0 0 38.1 1.5 24 0.94 0.11 0.107 0.097 0.03 0.044 0.154 0.034 0.169 0.1 0.038 0.25 0.08 0.059 0.07 0.399 0.844 0.02 0.073 0.1 0.162 4.63 0.607 0.928 1.8 0.058 0.991 0.191 1.06 0.5 0.084 0.252 0.197 15 0.934 0.967 0.283 0.638 0.17 0.314 0.178 147 0.02 0.109 0.424 0.042 0.178 0.047 0.042 0 0 0.13 0.7 1.69 0 0 0.357 719 28129 509 654 447 2600 126 0.34 0.4 2.27 0 8 0 259 4 7527 Page 2 of 5 5.3 Nutrient 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 11 12 13 17 18 25 23 2.09 884 0 100 14 15 16 38 130 448.8 149 60 1.16 2.69 0 25.56 10.51 0.16 0.28 50.1 4.42 0.1 0 5.57 2.9 0.85 0.42 0.26 0.33 3.75 4.65 4.64 5 0 8.63 28.17 2.5 20 0.38 0.22 0.18 10 0 17 20 136 5 1.24 1.2 0 0.99 2.38 3.57 0.45 10 0.01 10 12 0 33 40 87 11 104 1.22 33 43 295 130 56 24 370 157 35 414 212 466 222 199 4 3 1 9 80 0.73 0.06 0.19 0.49 86 5586 70 0.71 0.43 0.76 0.09 0.099 0.135 0.174 0.074 0.69 0.492 0 0.06 0.069 0.023 3.5 0 0 0 0 6.4 0 0.042 0.163 0.102 0 0.449 4.116 2.2 0.097 0 0.02 0.013 0 0.148 1.476 0 0 0.106 0.39 0 0 0.116 0.093 0 9.8 19.1 0.6 0.196 0.059 0.095 0.059 0.041 0.073 0.152 0.236 0.048 0.24 8.526 3.951 0.49 0.628 0.116 0.865 0.376 0.145 0.247 0.581 0.231 0.2 0.242 0.08 3.46 0 0 0 0.048 0 0 0 0 0.39 0 0 0 0 136 8190 623 409 Table 2: Nutritional Values of Various Foods per 100 g of Foodstuff Thiamine mg Riboflavin mg Niacin mg Pantothenic Acid mg Vitamin B6 mg Vitamin B12 mcg Vitamin A IU 0 0.04 0 0 DOR 0 7.5 0.5 Copyright 2012 Pearson Education, Inc. Publishing as Addison Wesley 19 20 21 201 3.5 11 23.6 128 0.09 14 105 Questions: Note: The MATLAB M-file diet.m contains data for the questions. Save the accompanying file diet.m as a text file and with the .m extension. Set the path browser in MATLAB to the folder where the file diet.m is located. Then type diet in the MATLAB command window for the data. 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 sauce. 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 in the M-file diet.m accompanying this file. 21 201 23 2.9 0.85 3.5 7 3.1 1.2 0.5 1.2 27 20 15 4 12 0 0 295 211 56 30 43 367 317 45 101 61.8 129 35 215 39 43 73 152 263 30 116 14 61.44 25 884 38 130 448.8 149 60 26 2.55 1.03 5.3 10.33 14.2 0.55 9.02 1.62 3.29 2.09 0 1.16 2.69 0 25.56 10.51 11.5 0.51 0.19 2.5 11.44 21.3 0.1 0.38 0.2 3.34 0.42 100 0.16 0.28 50.1 4.42 0.1 0.26 0.33 11 0 8.6 10.1 7.1 1.04 4.09 7.68 20.14 2.37 4.66 4.65 0 8.63 28.17 2.5 0 5.57 3.75 4.64 23.6 36 27 10 42 492.5 11 19 36 119.4 5 0.18 20 10 0 17 20 136 5 128 0.6 1.19 0.65 0.12 3.33 1.1 0.05 1.24 0.38 0.22 0 0.99 2.38 3.57 0.45 0.09 9 19.2 8 36 6 13.44 10 0.01 10 44 148 337 9 180 45 93.4 104 1.22 33 323 369 290 151.5 370 0 157 69 21 35 354 270 1116 0 2 8 49 4 0.04 3 1 5290 0.33 0.2 0.4 0.92 2.88 0.07 1.27 0.25 0.38 0.73 0.06 0.19 0.49 0.143 0.083 0.047 0.1 0.013 0.032 0.044 0.251 0.037 0.01 0.492 0 0.06 0.069 0 62 9.3 000 38.1 1.5 24 0.94 3.5 0 6.4 0.11 0.107 0.097 0.03 0.044 0.154 0.034 0.169 0.1 0.038 0.102 0 0.042 0.163 0.25 0.08 0.059 0.07 0.399 0.844 0.02 0.073 0.1 0.162 0.449 0 0.02 0.013 4.63 0.607 0.928 1.8 0.058 0.991 0.191 1.06 0.5 0.084 4.116 0 0.148 1.476 0.34 0.252 0.197 15 0.934 0.967 0.283 0.638 0.17 0.314 2.2 0 0.106 0.39 0.4 0.178 147 0.02 0.109 0.424 0.042 0.178 0.047 0.042 0.097 0 0.116 0.093 2.27 0 0 0.13 0.7 1.69 000 0.357 0 0 0 0 0 719 28129 509 654 447 259 8 2600 126 0 0 0 0 0 33 40 87 11 14 130 56 24 105 7.5 414 212 466 222 199 0.5 86 5586 70 9 80 0 0.71 0.43 0.76 0.09 0.69 0 0.099 0.135 0.174 0.074 0.023 0 0 0 9.8 19.1 0.6 0 0.196 0.059 0.095 0.059 0.041 0 0.073 0.152 0.236 0.048 0.24 0 8.526 3.951 0.49 0.628 0.116 0 0.865 0.376 0.145 0.247 0.581 0 0.231 0.20.242 0.08 0.048 0 3.46 00 0 0.39 0 136 0 8190 623 409 Application Project: Diet Problems 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 v which is in the M-file diet.m. 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? Nutrient Calories Protein Fat Carbohydrates Calcium Iron Magnesium Phosphorus Potassium Sodium Zinc Copper Vitamin C Thiamine Riboflavin Niacin Pantothenic Acid Vitamin B6 Vitamin B12 Vitamin A Amount 8279.12 kcal 608.81 g 387.6 g 604.48 g 4067.42 mg 93.34 mg 1714.73 mg 8488.03 mg 18023.48 mg 8846.38 mg 36009.75 mg 6.67 mcg 604.06 mg 6.77 mg 10.61 mg 28212.10 mg 103.11 mg Page 4 of 5 189.81 mg 51.78 mcg 95382.93 IU Table 3: Sumo Wrestler Diet v = [8279.12, 608.81, 387.60, 604.48, 4067.42, 93.34, 1714.73, 8488.03, 18023.48, 8846.38, 36009.75, 6.67, 604.06, 6.77, 10.61, 28212.10, 103.11, 189.81, 51.78, 95382.93]'; 5. The United States Food and Drug Administration (FDA) provides Recommended Daily Values for use on food labels. Table 4 gives the FDA's recommendations, which are also stored in the vector v2 which is in the accompanying M-file diet.m. Is it possible to combine the foods from the table to approximate these nutritional values? Nutrient Calories Protein Fat Carbohydrates Calcium Iron Magnesium Phosphorus Potassium Sodium Zinc Copper Vitamin C Thiamine Riboflavin Niacin Pantothenic Acid Vitamin B6 Vitamin B12 Vitamin A Amount 2000 kcal 50 g 65 g 300 g 1000 mg 18 mg 400 mg 1000 mg 3500 mg 2400 mg 15 mg 2000 mcg 60 mg 2 mg 2 mg 20 mg 10 mg 2 mg 6 mcg 5000 IU Table 4: FDA Recommended Daily Values v2 = [2000 50 65 300 1000 18 400 1000 3500 2400 15 20 10 2 6 5000 ]'; 2000 60 2 2 Reference: U.S. Department of Agriculture, Agricultural Research Service. 2001. USDA Nutrient Database for Standard Reference, Release 14. Nutrient Data Laboratory Home Page, www.ars.usda.gov/Services/docs.htm?docid=8964. To look up calories or nutrients for a particular food, use www.nal.usda.gov/fnic/foodcomp/search/.