Milk Collection Zone: Circle, Chord and the Cost of Extending the Radius
The Erode district milk cooperative is redrawing its collection zone on a coordinate map where one unit is one kilometre. The proposed zone is a circle with centre C(5, -2) and radius 10 km. Two villages appear on the map: P(13, 4), which already supplies the cooperative, and Q(-4, 6), which is served by a private trader and could be won over. A new state highway runs along the line x - 2y + 3 = 0, and the cooperative's tankers must cross the zone along this road, so the transport clerk needs the length of that crossing. Extending the circle is costed at Rs 35,000 per additional kilometre of radius per year, covering fencing, chilling capacity and route changes. Village Q would bring about 900 litres a day at a net margin of Rs 1.20 per litre. The board meets next week, and the manager has been asked for a recommendation that shows the numbers rather than a preference.
1. Determine whether village P and village Q lie inside, on, or outside the proposed zone, showing the distance calculation for each.
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Use the distance from the centre C(5, -2).
For P(13, 4): d = square root of [(13 - 5)^2 + (4 + 2)^2] = square root of (64 + 36) = square root of 100 = 10 km. This equals the radius exactly, so P lies on the boundary and is already covered.
For Q(-4, 6): d = square root of [(-4 - 5)^2 + (6 + 2)^2] = square root of (81 + 64) = square root of 145 = 12.04 km (approximately). Since 12.04 > 10, Q lies outside the zone, short of coverage by about 2.04 km.
So the present circle covers P on its rim and misses Q; the decision problem is entirely about Q. Note that the answer is not symmetric: a village on the rim is served, but one just over 2 km beyond is not, and the required extension is measured as the difference of the two distances, 12.04 - 10 = 2.04 km, not as any straight-line distance between P and Q.
2. Find the length of the highway, given by x - 2y + 3 = 0, that lies inside the collection zone.
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The chord length depends on the perpendicular distance from the centre to the line, which is the distance from C(5, -2) to x - 2y + 3 = 0.
d = |5 - 2(-2) + 3| divided by square root of (1^2 + (-2)^2) = |5 + 4 + 3| / square root of 5 = 12 / 2.2361 = 5.367 km.
Since 5.367 < 10, the road cuts the circle in two points. For a chord at distance d from the centre, half the chord is square root of (r^2 - d^2) = square root of (100 - 28.8) = square root of 71.2 = 8.438 km.
Chord length = 2 x 8.438 = 16.88 km approximately.
Interpretation for the clerk: the tanker route along the highway stays inside the zone for about 16.9 km, which is the stretch over which pickups can be planned on that road. If the zone were extended to reach Q, the perpendicular distance to the same line would change and the chord would have to be recomputed.
3. Advise the board whether to extend the radius so that village Q is included. Support the recommendation with an annual cost-benefit comparison.
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Required radius to include Q = distance CQ = 12.04 km, so the extension is 12.04 - 10 = 2.04 km.
Annual cost of extension = 2.04 x Rs 35,000 = Rs 71,400 (about Rs 71,300 if the unrounded 2.038 is used).
Annual benefit from Q = 900 litres per day x Rs 1.20 per litre x 365 days = Rs 3,94,200.
Net annual gain = 3,94,200 - 71,400 = Rs 3,22,800, a benefit-cost ratio of about 5.5 to 1. The recommendation is to extend the radius to about 12.1 km, which also gives a small buffer against mapping error.
Caveats to place before the board: the benefit assumes Q's 900 litres are actually captured and that the margin of Rs 1.20 holds; if a price war with the private trader halves the margin, the gain still exceeds the cost. Extending to 12.1 km may also pull in unlisted households, which would strengthen the case. The decision is not close, so the board should treat it as a capacity and route-planning issue rather than a financial one.