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 costs Rs 12 per gram. Food Y contains 8 units of Vitamin D per gram and 12 units of Vitamin E per gram, and costs Rs 20 per gram. The daily minimum requirements of Vitamin D and Vitamin E are 100 units and 120 units respectively.
Suppose \(x\) is the quantity (in grams) of food X, and \(y\) is the quantity (in grams) of food Y.
Answer the following question based on the paragraph above.
Which of the following are the quantities (in grams) of food X and food Y respectively when the cost of food is minimum?
- A.
0 and
\(12 \frac 1 2\) - B.
15 and
\( \frac 5 4\) - C.
\(\frac {120} {7}\)and 0 - D.
0 and 10
Attempted by 8 students.
Show answer & explanation
Correct answer: B
Concept. This is a cost-minimisation linear programming problem. The Fundamental Theorem of Linear Programming says that the optimum of a linear objective over a region bounded by linear constraints always occurs at a corner (vertex) of the feasible region. So the method is: list the constraints, find the corner points, evaluate the objective at each, and pick the corner that minimises it.
Set up. Let the objective be the total cost C = 12x + 20y, subject to:
Vitamin D: 6x + 8y ≥ 100
Vitamin E: 7x + 12y ≥ 120
Non-negativity: x ≥ 0, y ≥ 0
Find the corner where the two requirement lines meet by solving them as equalities:
From 6x + 8y = 100, express x: x = (100 − 8y)/6 = 50/3 − (4/3)y.
Substitute into 7x + 12y = 120: 7(50/3 − (4/3)y) + 12y = 120.
Simplify: 350/3 − (28/3)y + 12y = 120 → 350/3 + (8/3)y = 120 → (8/3)y = 10/3 → y = 5/4.
Back-substitute: x = 50/3 − (4/3)(5/4) = 50/3 − 5/3 = 45/3 = 15.
So this corner is (x, y) = (15, 5/4).
Evaluate the cost at every feasible corner:
Corner (x, y) | Feasible? | Cost C = 12x + 20y |
|---|---|---|
(0, 25/2) | Yes | Rs 250 |
(120/7, 0) | Yes | Rs 1440/7 ≈ 205.71 |
(15, 5/4) | Yes | Rs 205 |
(0, 10) | No (Vit D = 80 < 100) | — |
Cross-check. The smallest cost among the feasible corners is Rs 205, at (15, 5/4). At that point both vitamin constraints hold with equality (6(15) + 8(5/4) = 100 and 7(15) + 12(5/4) = 120), which confirms it is a genuine vertex of the feasible region, so by the corner-point theorem it is the global minimum.
Result. The cost is minimised when x = 15 g of food X and y = 5/4 g of food Y, giving a minimum cost of Rs 205.