Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per…
2023
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram.Food Y contains 8 units of Vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of Vitamin D and E are 100 units and 120 units respectively.
Suppose \(x\) is quantity (in gram) of food X, \(y\) is quantity (in gram) of food Y.
Answering the following question based on the above paragraph given.
The minimum cost of food is :
- A.
205
- B.
250
- C.
330
- D.
200
Attempted by 10 students.
Show answer & explanation
Correct answer: A
Set up:
Let x be grams of food X and y be grams of food Y. Minimize the cost C = 12x + 20y subject to the constraints:
Vitamin D: 6x + 8y ≥ 100
Vitamin E: 7x + 12y ≥ 120
Non‑negativity: x ≥ 0, y ≥ 0
Solve by checking corner points of the feasible region. First find the intersection of the two nutrient equalities.
Solve 6x + 8y = 100 and 7x + 12y = 120. Multiply the first equation by 3: 18x + 24y = 300. Multiply the second by 2: 14x + 24y = 240. Subtract to get 4x = 60, so x = 15.
Substitute x = 15 into 6x + 8y = 100: 90 + 8y = 100 ⇒ 8y = 10 ⇒ y = 1.25.
Compute the cost at this intersection:
C = 12(15) + 20(1.25) = 180 + 25 = 205.
Check other boundary points to confirm this is the minimum:
If x = 0 then y must be at least 100/8 = 12.5 to meet Vitamin D, giving cost 20 × 12.5 = 250.
If y = 0 then to meet Vitamin E you need x ≥ 120/7 ≈ 17.1429, giving cost ≈ 12 × 17.1429 ≈ 205.71.
Conclusion: The minimum cost is 205 achieved at x = 15 g of food X and y = 1.25 g of food Y.