The total cost, $\(C\), of manufacturing \(x\) batches of biscuit boxes is given by the formula \(C=ax+b\), where \(a\) and \(b\) are constants. When 200 batches of biscuit boxes are manufactured, the total cost is $55000. When 500 batches of biscuit boxes are manufactured, the total cost is $62500. Find the value of \(b\).

For a similar problem on manufacturing costs, you can try the question Factory Equipment Dilemma.

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