Question

Maggie Stewart loves desserts. But, due to weight and cholesterol concerns, she has decided that she must plan her desserts carefully. There are two possible desserts she is considering: snack bars and icecream. After reading the nutrition labels on the snack bar and ice-cream packages, she learns that each serving of a snack bar weighs 37 grams and contains 120 calories and 5 grams of fat. Each serving of icecream weighs 65 grams and contains 160 calories and 10 grams of fat. Maggie will allow herself no more than 450 calories and 25 grams of fat in her daily desserts, but because she loves desserts so much, she requires at least 120 grams of dessert per day.

Answer 1

Answer:

The option that maximizes Maggie's taste index is 1 snack bar and 2 ice creams

Explanation:

snack bar                  ice cream

37 grams                   65 grams

120 calories              160 calories

5 grams of fat           10 grams of fat

Maggie wants to consume up to 450 calories and 25 grams of fat, but she needs at least 120 grams of dessert per day. Ice cream taste 95, snack bars 85.

• maximize taste index = [85(37X) + 95(65Y)] / (37X + 65Y)
• 5X + 10Y ≤ 25 ⇒ CONSTRAINT 1
• 120X + 160Y ≤ 450 ⇒ CONSTRAINT 2
• 37X + 65Y ≥ 120 ⇒ CONSTRAINT 3
• X ≥ 0 ⇒ CONSTRAINT 4
• Y ≥ 0 ⇒ CONSTRAINT 5

maximum possible combinations following constraint 1, 4 AND 5:

• option 1: 1 snack bar - 2 ice creams (5 + 20 = 25)
• option 2: 2 snack bars - 1 ice cream (10 + 10 = 20)
• option 3: 3 snack bars - 1 ice cream (15 + 10 = 25)

possible combinations following constraint 2:

• option 1: 1 snack bar - 2 ice creams (120 + 320 = 440)
• option 2: 2 snack bars - 1 ice cream (240 + 160 = 400)

possible combination following constraint 3:

• option 1: 1 snack bar - 2 ice creams (37 + 130 = 167)
• option 2: 2 snack bars - 1 ice cream (74 + 65 = 139)

since we only have two possibilities, we can calculate which one generates the highest taste index

maximize taste index = [85(37X) + 95(65Y)] / (37X + 65Y)

• option 1: 1 snack bar - 2 ice creams = [85(37) + 95(130)] / (37 + 130) = (3,145 + 12,350) / 167 = 92.78
• option 2: 2 snack bars - 1 ice cream = [85(74) + 95(65)] / (74 + 65) = (6,290 + 6,175) / 139 = 89.68