Robert Heun, president of Concrete Always, agrees to construct a concrete cart path at Dakota Golf Club. Concrete Always enters into a contract with Dakota to construct the path for $$188,000.$ In addition, as part of the contract, a performance bonus of $$37,600$ will be paid based on the timing of completion. The performance bonus will be paid fully if completed by the agreed$-$upon date. The performance bonus decreases by $$9,400$ per week for every week beyond the agreed$-$upon completion date. Robert has been involved in a number of contracts that had performance bonuses as part of the agreement in the past. As a result, he is fairly confident that he will receive a good portion of the performance bonus. Robert estimates, given the constraints of his schedule related to other jobs, that there is $55\%$ probability that he will complete the project on time, a $30\%$ probability that he will be $1$ week late, and a $15\%$ probability that he will be $2$ weeks late.

$($a$)$

Determine the transaction price that Concrete Always should compute for this agreement.

Transaction price $