Linear Graphs in Real Life Problems

Cash on hand of a company refers to the money a company can access to fund their business activities and operating cost. When a company earns money, the money is stored as cash on hand. Originally, the company earns a fixed amount of money per day. On the \(5\)th day, the company invests in a new facility that improves the product efficiency, which leads to a higher fixed amount of money earned per day afterwards. The cash on hand-time graph below shows the cash on hand of this company. Assume there are no other operating costs incurred in the \(10\) days of operation.

1. Find the cash on hand (in dollars) of this company at the beginning of this month.
2. Find the amount of money (in dollars) the company
   1. earned before purchasing the new facility,
   2. spent on purchasing the new facility,
   3. earned after purchasing the new facility, until the \(10\)th day.
3. Find the gradient of each of the following line segments, and state clearly what each gradient represents.
   1. \(AB\),
   2. \(CD\).
4. From the graph, which day does the company first recover the cost of the new facility? Round your answer to the nearest integer.