"A gem's price is proportional to the square of its weight."
\(P = k × W^2\), where W is the stone's weight and P is its price.
The cost of the uncut stone is \(18^2 × k = 324 k.\)
“Breaking the stone into four pieces, each with a unique integer weight, results in a maximum difference of 288000 between the highest and lowest possible total prices of the four pieces.”
The minimum profit is achieved when the fragmented stone weights are similar, specifically 3, 4, 5, and 6 units.
In this scenario, the aggregate value of the four stones is \((3^2+4^2+5^2+6^2)k=86k\).
The maximum profit is realized when the broken stone weights are disparate, for example, 1, 2, 3, and 12 units.
The aggregate value of the four stones in this scenario equals \((1^2+2^2+3^2+12^2)k=158k.\)
The total value difference is 2,88,000.
158k – 86k = 72k = 288000
This implies k = 4000.
Therefore, the original stone cost 324 k, which is 324 * 4000 = 12,96,000.
The correct option is (B): 1296000