A hostel mess mixes two foods P and Q. One kg of P gives 2 units of vitamin A and 1 unit of vitamin B; one kg of Q gives 1 unit of A and 2 units of B. The mix must give at least 8 units of A and 10 units of B. P costs ₹ 50 per kg and Q costs ₹ 70 per kg (example prices). Find the cheapest mix.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Let x kg of P and y kg of Q be used. Minimise Z = 50x + 70y subject to 2x + y ≥ 8, x + 2y ≥ 10, x ≥ 0, y ≥ 0.
2 · Draw the figure or sketch.
2x + y = 8 through (4, 0) and (0, 8)
x + 2y = 10 through (10, 0) and (0, 5)
O fails both, so shade away from O: the region is unbounded
3 · Write the method. Corner point method, with the half-plane test because the region is unbounded.
4 · Find the corners and the values of Z.
A(0, 8) and C(10, 0)
B: solve 2x + y = 8 and x + 2y = 10 to get (2, 4)
Z at A = 560, at B = 380, at C = 500
5 · Do the half-plane test.
Draw the dotted line 50x + 70y = 380
The half-plane 50x + 70y < 380 has no point in common with the region
6 · Say what it means. So 2 kg of P and 4 kg of Q meet both vitamin needs at the lowest cost.
7 · Check and box the answer. B(2, 4): 2(2) + 4 = 8 ≥ 8 and 2 + 8 = 10 ≥ 10. Minimum cost ₹ 380.
Scoring tip: For an unbounded region, the half-plane test is a separate mark. Never skip it.
